Main:z from

Percentage Accurate: 91.6% → 99.0%

Time: 28.4s

Alternatives: 24

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := z + \left(1 + z\right)\\ t_2 := \sqrt{\frac{1}{y}}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \sqrt{\frac{1}{z}}\\ t_6 := \sqrt{1 + y} - \sqrt{y}\\ t_7 := \left(t\_4 - \sqrt{x}\right) + t\_6\\ \mathbf{if}\;t\_7 \leq 10^{-5}:\\ \;\;\;\;t\_3 + 0.5 \cdot \left(t\_5 + \left(\sqrt{\frac{1}{x}} + t\_2\right)\right)\\ \mathbf{elif}\;t\_7 \leq 1.00005:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, t\_5 + t\_2, t\_4\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(t\_6 + \left(1 - \sqrt{x}\right)\right) + \frac{t\_1}{t\_1 \cdot \left(\sqrt{z} + \sqrt{1 + z}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ z (+ 1.0 z)))
        (t_2 (sqrt (/ 1.0 y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (sqrt (/ 1.0 z)))
        (t_6 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_7 (+ (- t_4 (sqrt x)) t_6)))
   (if (<= t_7 1e-5)
     (+ t_3 (* 0.5 (+ t_5 (+ (sqrt (/ 1.0 x)) t_2))))
     (if (<= t_7 1.00005)
       (+ t_3 (- (fma 0.5 (+ t_5 t_2) t_4) (sqrt x)))
       (+
        t_3
        (+
         (+ t_6 (- 1.0 (sqrt x)))
         (/ t_1 (* t_1 (+ (sqrt z) (sqrt (+ 1.0 z)))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = z + (1.0 + z);
	double t_2 = sqrt((1.0 / y));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double t_5 = sqrt((1.0 / z));
	double t_6 = sqrt((1.0 + y)) - sqrt(y);
	double t_7 = (t_4 - sqrt(x)) + t_6;
	double tmp;
	if (t_7 <= 1e-5) {
		tmp = t_3 + (0.5 * (t_5 + (sqrt((1.0 / x)) + t_2)));
	} else if (t_7 <= 1.00005) {
		tmp = t_3 + (fma(0.5, (t_5 + t_2), t_4) - sqrt(x));
	} else {
		tmp = t_3 + ((t_6 + (1.0 - sqrt(x))) + (t_1 / (t_1 * (sqrt(z) + sqrt((1.0 + z))))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(z + Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 / y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = sqrt(Float64(1.0 / z))
	t_6 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_7 = Float64(Float64(t_4 - sqrt(x)) + t_6)
	tmp = 0.0
	if (t_7 <= 1e-5)
		tmp = Float64(t_3 + Float64(0.5 * Float64(t_5 + Float64(sqrt(Float64(1.0 / x)) + t_2))));
	elseif (t_7 <= 1.00005)
		tmp = Float64(t_3 + Float64(fma(0.5, Float64(t_5 + t_2), t_4) - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(Float64(t_6 + Float64(1.0 - sqrt(x))) + Float64(t_1 / Float64(t_1 * Float64(sqrt(z) + sqrt(Float64(1.0 + z)))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z + N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]}, If[LessEqual[t$95$7, 1e-5], N[(t$95$3 + N[(0.5 * N[(t$95$5 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.00005], N[(t$95$3 + N[(N[(0.5 * N[(t$95$5 + t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(t$95$6 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / N[(t$95$1 * N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00000000000000008e-5

   1. Initial program 73.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 82.1

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 82.1%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 93.2

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 93.2%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. /-lowering-/.f64 50.5

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 50.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000008e-5 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00005000000000011

   1. Initial program 95.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. sqrt-lowering-sqrt.f64 32.2

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 32.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 1.00005000000000011 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 93.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 93.9%

      \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(z + \left(1 + z\right)\right) \cdot 1}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(z + \left(1 + z\right)\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.00005:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{z + \left(1 + z\right)}{\left(z + \left(1 + z\right)\right) \cdot \left(\sqrt{z} + \sqrt{1 + z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + z}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_1 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right)\right)\\ t_7 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_6 \leq 0.1:\\ \;\;\;\;t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_7\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3 + t\_7, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_6 \leq 2.0002:\\ \;\;\;\;t\_5 + \left(\mathsf{fma}\left(0.5, t\_3, t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + t\_4\right) + \left(t\_5 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6
         (+ t_1 (+ (- t_4 (sqrt z)) (+ (- t_5 (sqrt x)) (- t_2 (sqrt y))))))
        (t_7 (sqrt (/ 1.0 y))))
   (if (<= t_6 0.1)
     (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 x)) t_7)))
     (if (<= t_6 1.00005)
       (- (fma 0.5 (+ t_3 t_7) 1.0) (sqrt x))
       (if (<= t_6 2.0002)
         (+ t_5 (- (fma 0.5 t_3 t_2) (+ (sqrt x) (sqrt y))))
         (+ (+ t_2 t_4) (- t_5 (+ (sqrt x) (+ (sqrt y) (sqrt z))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + z));
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_1 + ((t_4 - sqrt(z)) + ((t_5 - sqrt(x)) + (t_2 - sqrt(y))));
	double t_7 = sqrt((1.0 / y));
	double tmp;
	if (t_6 <= 0.1) {
		tmp = t_1 + (0.5 * (sqrt((1.0 / x)) + t_7));
	} else if (t_6 <= 1.00005) {
		tmp = fma(0.5, (t_3 + t_7), 1.0) - sqrt(x);
	} else if (t_6 <= 2.0002) {
		tmp = t_5 + (fma(0.5, t_3, t_2) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (t_2 + t_4) + (t_5 - (sqrt(x) + (sqrt(y) + sqrt(z))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_1 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_5 - sqrt(x)) + Float64(t_2 - sqrt(y)))))
	t_7 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_6 <= 0.1)
		tmp = Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_7)));
	elseif (t_6 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(t_3 + t_7), 1.0) - sqrt(x));
	elseif (t_6 <= 2.0002)
		tmp = Float64(t_5 + Float64(fma(0.5, t_3, t_2) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(t_2 + t_4) + Float64(t_5 - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 0.1], N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(0.5 * N[(t$95$3 + t$95$7), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0002], N[(t$95$5 + N[(N[(0.5 * t$95$3 + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + t$95$4), $MachinePrecision] + N[(t$95$5 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.10000000000000001

   1. Initial program 13.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 37.5

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 37.5%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 74.7

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 74.7%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 74.7

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 74.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00005000000000011

   1. Initial program 94.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 24.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 24.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 25.3

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 25.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 25.8

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 25.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 25.8

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 25.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 37.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 26.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.00019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 33.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 33.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + z}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_1 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right)\right)\\ t_7 := \sqrt{\frac{1}{y}}\\ t_8 := \sqrt{x} + \sqrt{y}\\ \mathbf{if}\;t\_6 \leq 0.1:\\ \;\;\;\;t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_7\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3 + t\_7, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_6 \leq 2.0002:\\ \;\;\;\;t\_5 + \left(\mathsf{fma}\left(0.5, t\_3, t\_2\right) - t\_8\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 + \left(1 + t\_2\right)\right) - \left(\sqrt{z} + t\_8\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6
         (+ t_1 (+ (- t_4 (sqrt z)) (+ (- t_5 (sqrt x)) (- t_2 (sqrt y))))))
        (t_7 (sqrt (/ 1.0 y)))
        (t_8 (+ (sqrt x) (sqrt y))))
   (if (<= t_6 0.1)
     (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 x)) t_7)))
     (if (<= t_6 1.00005)
       (- (fma 0.5 (+ t_3 t_7) 1.0) (sqrt x))
       (if (<= t_6 2.0002)
         (+ t_5 (- (fma 0.5 t_3 t_2) t_8))
         (- (+ t_4 (+ 1.0 t_2)) (+ (sqrt z) t_8)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + z));
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_1 + ((t_4 - sqrt(z)) + ((t_5 - sqrt(x)) + (t_2 - sqrt(y))));
	double t_7 = sqrt((1.0 / y));
	double t_8 = sqrt(x) + sqrt(y);
	double tmp;
	if (t_6 <= 0.1) {
		tmp = t_1 + (0.5 * (sqrt((1.0 / x)) + t_7));
	} else if (t_6 <= 1.00005) {
		tmp = fma(0.5, (t_3 + t_7), 1.0) - sqrt(x);
	} else if (t_6 <= 2.0002) {
		tmp = t_5 + (fma(0.5, t_3, t_2) - t_8);
	} else {
		tmp = (t_4 + (1.0 + t_2)) - (sqrt(z) + t_8);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_1 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_5 - sqrt(x)) + Float64(t_2 - sqrt(y)))))
	t_7 = sqrt(Float64(1.0 / y))
	t_8 = Float64(sqrt(x) + sqrt(y))
	tmp = 0.0
	if (t_6 <= 0.1)
		tmp = Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_7)));
	elseif (t_6 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(t_3 + t_7), 1.0) - sqrt(x));
	elseif (t_6 <= 2.0002)
		tmp = Float64(t_5 + Float64(fma(0.5, t_3, t_2) - t_8));
	else
		tmp = Float64(Float64(t_4 + Float64(1.0 + t_2)) - Float64(sqrt(z) + t_8));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.1], N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(0.5 * N[(t$95$3 + t$95$7), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0002], N[(t$95$5 + N[(N[(0.5 * t$95$3 + t$95$2), $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.10000000000000001

   1. Initial program 13.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 37.5

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 37.5%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 74.7

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 74.7%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 74.7

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 74.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00005000000000011

   1. Initial program 94.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 24.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 24.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 25.3

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 25.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 25.8

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 25.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 25.8

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 25.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 37.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 26.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.00019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 74.0

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 74.0%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right) \]

      14. sqrt-lowering-sqrt.f64 27.2

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 27.2%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \sqrt{1 + y}\\ t_6 := t\_2 + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_5 - \sqrt{y}\right)\right)\\ t_7 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_6 \leq 0.1:\\ \;\;\;\;t\_3 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_7\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + t\_7, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_6 \leq 2.5:\\ \;\;\;\;t\_4 + \left(\left(t\_5 + \frac{1}{\sqrt{z} + t\_1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_2 + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ t_2 (+ (- t_4 (sqrt x)) (- t_5 (sqrt y)))))
        (t_7 (sqrt (/ 1.0 y))))
   (if (<= t_6 0.1)
     (+ t_3 (* 0.5 (+ (sqrt (/ 1.0 x)) t_7)))
     (if (<= t_6 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) t_7) 1.0) (sqrt x))
       (if (<= t_6 2.5)
         (+ t_4 (- (+ t_5 (/ 1.0 (+ (sqrt z) t_1))) (+ (sqrt x) (sqrt y))))
         (+ t_3 (+ t_2 (- (- 2.0 (sqrt x)) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_2 + ((t_4 - sqrt(x)) + (t_5 - sqrt(y)));
	double t_7 = sqrt((1.0 / y));
	double tmp;
	if (t_6 <= 0.1) {
		tmp = t_3 + (0.5 * (sqrt((1.0 / x)) + t_7));
	} else if (t_6 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + t_7), 1.0) - sqrt(x);
	} else if (t_6 <= 2.5) {
		tmp = t_4 + ((t_5 + (1.0 / (sqrt(z) + t_1))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_3 + (t_2 + ((2.0 - sqrt(x)) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_2 + Float64(Float64(t_4 - sqrt(x)) + Float64(t_5 - sqrt(y))))
	t_7 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_6 <= 0.1)
		tmp = Float64(t_3 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_7)));
	elseif (t_6 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + t_7), 1.0) - sqrt(x));
	elseif (t_6 <= 2.5)
		tmp = Float64(t_4 + Float64(Float64(t_5 + Float64(1.0 / Float64(sqrt(z) + t_1))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_3 + Float64(t_2 + Float64(Float64(2.0 - sqrt(x)) - sqrt(y))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 0.1], N[(t$95$3 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.5], N[(t$95$4 + N[(N[(t$95$5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$2 + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.10000000000000001

   1. Initial program 54.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 65.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 65.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 84.1

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 84.1%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 84.0

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 84.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 35.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 35.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 37.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 37.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 39.1

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 22.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 22.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z}} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   7. Simplified 29.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 97.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 - \sqrt{x}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(2 - \color{blue}{\sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. sqrt-lowering-sqrt.f64 97.5

         \[\leadsto \left(\left(\left(2 - \sqrt{x}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 97.5%

      \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + x}\\ t_5 := \sqrt{1 + y}\\ t_6 := t\_1 + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_5 - \sqrt{y}\right)\right)\\ t_7 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_6 \leq 0.1:\\ \;\;\;\;t\_2 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_7\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3 + t\_7, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_6 \leq 2.0002:\\ \;\;\;\;t\_4 + \left(\mathsf{fma}\left(0.5, t\_3, t\_5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ t_1 (+ (- t_4 (sqrt x)) (- t_5 (sqrt y)))))
        (t_7 (sqrt (/ 1.0 y))))
   (if (<= t_6 0.1)
     (+ t_2 (* 0.5 (+ (sqrt (/ 1.0 x)) t_7)))
     (if (<= t_6 1.00005)
       (- (fma 0.5 (+ t_3 t_7) 1.0) (sqrt x))
       (if (<= t_6 2.0002)
         (+ t_4 (- (fma 0.5 t_3 t_5) (+ (sqrt x) (sqrt y))))
         (+ t_2 (+ t_1 (- (- 2.0 (sqrt x)) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + x));
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_1 + ((t_4 - sqrt(x)) + (t_5 - sqrt(y)));
	double t_7 = sqrt((1.0 / y));
	double tmp;
	if (t_6 <= 0.1) {
		tmp = t_2 + (0.5 * (sqrt((1.0 / x)) + t_7));
	} else if (t_6 <= 1.00005) {
		tmp = fma(0.5, (t_3 + t_7), 1.0) - sqrt(x);
	} else if (t_6 <= 2.0002) {
		tmp = t_4 + (fma(0.5, t_3, t_5) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_2 + (t_1 + ((2.0 - sqrt(x)) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_1 + Float64(Float64(t_4 - sqrt(x)) + Float64(t_5 - sqrt(y))))
	t_7 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_6 <= 0.1)
		tmp = Float64(t_2 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_7)));
	elseif (t_6 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(t_3 + t_7), 1.0) - sqrt(x));
	elseif (t_6 <= 2.0002)
		tmp = Float64(t_4 + Float64(fma(0.5, t_3, t_5) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(Float64(2.0 - sqrt(x)) - sqrt(y))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 0.1], N[(t$95$2 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(0.5 * N[(t$95$3 + t$95$7), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0002], N[(t$95$4 + N[(N[(0.5 * t$95$3 + t$95$5), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.10000000000000001

   1. Initial program 54.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 65.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 65.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 84.1

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 84.1%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 84.0

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 84.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 35.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 35.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 37.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 37.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 39.1

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 22.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 22.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00019999999999998

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 25.7

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.00019999999999998 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 91.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 91.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 - \sqrt{x}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(2 - \color{blue}{\sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. sqrt-lowering-sqrt.f64 88.5

         \[\leadsto \left(\left(\left(2 - \sqrt{x}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 88.5%

      \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.0002:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \left(t\_1 - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\\ t_5 := \sqrt{\frac{1}{y}}\\ t_6 := 1 + t\_3\\ \mathbf{if}\;t\_4 \leq 0.1:\\ \;\;\;\;t\_2 + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_5\right)\\ \mathbf{elif}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + t\_5, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_2 + \left(\left(t\_6 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + t\_6\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4
         (+
          (- t_1 (sqrt z))
          (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_3 (sqrt y)))))
        (t_5 (sqrt (/ 1.0 y)))
        (t_6 (+ 1.0 t_3)))
   (if (<= t_4 0.1)
     (+ t_2 (* 0.5 (+ (sqrt (/ 1.0 x)) t_5)))
     (if (<= t_4 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) t_5) 1.0) (sqrt x))
       (if (<= t_4 2.0)
         (+ t_2 (- (- t_6 (sqrt y)) (sqrt x)))
         (- (+ t_1 t_6) (+ (sqrt z) (+ (sqrt x) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + y));
	double t_4 = (t_1 - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_3 - sqrt(y)));
	double t_5 = sqrt((1.0 / y));
	double t_6 = 1.0 + t_3;
	double tmp;
	if (t_4 <= 0.1) {
		tmp = t_2 + (0.5 * (sqrt((1.0 / x)) + t_5));
	} else if (t_4 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + t_5), 1.0) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = t_2 + ((t_6 - sqrt(y)) - sqrt(x));
	} else {
		tmp = (t_1 + t_6) - (sqrt(z) + (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_3 - sqrt(y))))
	t_5 = sqrt(Float64(1.0 / y))
	t_6 = Float64(1.0 + t_3)
	tmp = 0.0
	if (t_4 <= 0.1)
		tmp = Float64(t_2 + Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_5)));
	elseif (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + t_5), 1.0) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(t_2 + Float64(Float64(t_6 - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(Float64(t_1 + t_6) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(1.0 + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 0.1], N[(t$95$2 + N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(t$95$2 + N[(N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.10000000000000001

   1. Initial program 54.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 65.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 65.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 84.1

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 84.1%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 84.0

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 84.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 35.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 35.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 37.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 37.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 39.1

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 22.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 22.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 64.6

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 64.6%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 33.7

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 33.7%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 33.4

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 88.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right) \]

      14. sqrt-lowering-sqrt.f64 61.9

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := 1 + t\_1\\ t_3 := \sqrt{1 + z}\\ t_4 := \left(t\_3 - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_2 - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + t\_2\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ 1.0 t_1))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4
         (+
          (- t_3 (sqrt z))
          (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_4 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 1.0) (sqrt x))
       (if (<= t_4 2.0)
         (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (- (- t_2 (sqrt y)) (sqrt x)))
         (- (+ t_3 t_2) (+ (sqrt z) (+ (sqrt x) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = 1.0 + t_1;
	double t_3 = sqrt((1.0 + z));
	double t_4 = (t_3 - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_4 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), 1.0) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((t_2 - sqrt(y)) - sqrt(x));
	} else {
		tmp = (t_3 + t_2) - (sqrt(z) + (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(1.0 + t_1)
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), 1.0) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_2 - sqrt(y)) - sqrt(x)));
	else
		tmp = Float64(Float64(t_3 + t_2) - Float64(sqrt(z) + Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + t$95$2), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.10000000000000001

   1. Initial program 54.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 65.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 65.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 19.6

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 35.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 35.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 37.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 37.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 39.1

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 22.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 22.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 64.6

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 64.6%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 33.7

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 33.7%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 33.4

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 88.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right) \]

      14. sqrt-lowering-sqrt.f64 61.9

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.9999999999999998:\\ \;\;\;\;t\_4 + \left(t\_3 + \frac{1}{\sqrt{x} + t\_5}\right)\\ \mathbf{elif}\;t\_6 \leq 2.5:\\ \;\;\;\;t\_5 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(t\_3 + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (+ t_3 (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_6 0.9999999999999998)
     (+ t_4 (+ t_3 (/ 1.0 (+ (sqrt x) t_5))))
     (if (<= t_6 2.5)
       (+ t_5 (- (+ t_1 (/ 1.0 (+ (sqrt z) t_2))) (+ (sqrt x) (sqrt y))))
       (+ t_4 (+ t_3 (- (- 2.0 (sqrt x)) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_6 <= 0.9999999999999998) {
		tmp = t_4 + (t_3 + (1.0 / (sqrt(x) + t_5)));
	} else if (t_6 <= 2.5) {
		tmp = t_5 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_4 + (t_3 + ((2.0 - sqrt(x)) - sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = sqrt((1.0d0 + x))
    t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)))
    if (t_6 <= 0.9999999999999998d0) then
        tmp = t_4 + (t_3 + (1.0d0 / (sqrt(x) + t_5)))
    else if (t_6 <= 2.5d0) then
        tmp = t_5 + ((t_1 + (1.0d0 / (sqrt(z) + t_2))) - (sqrt(x) + sqrt(y)))
    else
        tmp = t_4 + (t_3 + ((2.0d0 - sqrt(x)) - sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - Math.sqrt(x)) + (t_1 - Math.sqrt(y)));
	double tmp;
	if (t_6 <= 0.9999999999999998) {
		tmp = t_4 + (t_3 + (1.0 / (Math.sqrt(x) + t_5)));
	} else if (t_6 <= 2.5) {
		tmp = t_5 + ((t_1 + (1.0 / (Math.sqrt(z) + t_2))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = t_4 + (t_3 + ((2.0 - Math.sqrt(x)) - Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((1.0 + x))
	t_6 = t_3 + ((t_5 - math.sqrt(x)) + (t_1 - math.sqrt(y)))
	tmp = 0
	if t_6 <= 0.9999999999999998:
		tmp = t_4 + (t_3 + (1.0 / (math.sqrt(x) + t_5)))
	elif t_6 <= 2.5:
		tmp = t_5 + ((t_1 + (1.0 / (math.sqrt(z) + t_2))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = t_4 + (t_3 + ((2.0 - math.sqrt(x)) - math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_6 <= 0.9999999999999998)
		tmp = Float64(t_4 + Float64(t_3 + Float64(1.0 / Float64(sqrt(x) + t_5))));
	elseif (t_6 <= 2.5)
		tmp = Float64(t_5 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_4 + Float64(t_3 + Float64(Float64(2.0 - sqrt(x)) - sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = sqrt((1.0 + x));
	t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)));
	tmp = 0.0;
	if (t_6 <= 0.9999999999999998)
		tmp = t_4 + (t_3 + (1.0 / (sqrt(x) + t_5)));
	elseif (t_6 <= 2.5)
		tmp = t_5 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(x) + sqrt(y)));
	else
		tmp = t_4 + (t_3 + ((2.0 - sqrt(x)) - sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.9999999999999998], N[(t$95$4 + N[(t$95$3 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.5], N[(t$95$5 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(t$95$3 + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.99999999999999978

   1. Initial program 61.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 66.0%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 73.7

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 73.7%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 70.5

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 70.5%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.99999999999999978 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.0

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.0%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{1 + z}} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   7. Simplified 26.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 97.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(2 - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 - \sqrt{x}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(2 - \color{blue}{\sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. sqrt-lowering-sqrt.f64 97.5

         \[\leadsto \left(\left(\left(2 - \sqrt{x}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 97.5%

      \[\leadsto \left(\color{blue}{\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.9999999999999998:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_2 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_2 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + t\_1\right) - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2
         (+
          (- (sqrt (+ 1.0 z)) (sqrt z))
          (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_2 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_2 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 1.0) (sqrt x))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (- (- (+ 1.0 t_1) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = (sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_2 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), 1.0) - sqrt(x);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (((1.0 + t_1) - sqrt(y)) - sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_2 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), 1.0) - sqrt(x));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(1.0 + t_1) - sqrt(y)) - sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.10000000000000001

   1. Initial program 54.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 65.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 65.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 19.6

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 94.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 35.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 35.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 37.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 37.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 39.1

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 22.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 22.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 70.4

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 70.4%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 30.0

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\sqrt{\sqrt{1 + y} \cdot \sqrt{1 + y}}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 29.8

          \[\leadsto \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied egg-rr 29.8%

       \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{y}}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{\frac{1}{z}}\\ t_5 := \sqrt{1 + y} - \sqrt{y}\\ t_6 := \left(t\_3 - \sqrt{x}\right) + t\_5\\ \mathbf{if}\;t\_6 \leq 10^{-5}:\\ \;\;\;\;t\_2 + 0.5 \cdot \left(t\_4 + \left(\sqrt{\frac{1}{x}} + t\_1\right)\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, t\_4 + t\_1, t\_3\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_5 + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (sqrt (/ 1.0 z)))
        (t_5 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_6 (+ (- t_3 (sqrt x)) t_5)))
   (if (<= t_6 1e-5)
     (+ t_2 (* 0.5 (+ t_4 (+ (sqrt (/ 1.0 x)) t_1))))
     (if (<= t_6 1.00005)
       (+ t_2 (- (fma 0.5 (+ t_4 t_1) t_3) (sqrt x)))
       (+
        t_2
        (+ (+ t_5 (- 1.0 (sqrt x))) (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 / z));
	double t_5 = sqrt((1.0 + y)) - sqrt(y);
	double t_6 = (t_3 - sqrt(x)) + t_5;
	double tmp;
	if (t_6 <= 1e-5) {
		tmp = t_2 + (0.5 * (t_4 + (sqrt((1.0 / x)) + t_1)));
	} else if (t_6 <= 1.00005) {
		tmp = t_2 + (fma(0.5, (t_4 + t_1), t_3) - sqrt(x));
	} else {
		tmp = t_2 + ((t_5 + (1.0 - sqrt(x))) + (1.0 / (sqrt(z) + sqrt((1.0 + z)))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = sqrt(Float64(1.0 / z))
	t_5 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_6 = Float64(Float64(t_3 - sqrt(x)) + t_5)
	tmp = 0.0
	if (t_6 <= 1e-5)
		tmp = Float64(t_2 + Float64(0.5 * Float64(t_4 + Float64(sqrt(Float64(1.0 / x)) + t_1))));
	elseif (t_6 <= 1.00005)
		tmp = Float64(t_2 + Float64(fma(0.5, Float64(t_4 + t_1), t_3) - sqrt(x)));
	else
		tmp = Float64(t_2 + Float64(Float64(t_5 + Float64(1.0 - sqrt(x))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, 1e-5], N[(t$95$2 + N[(0.5 * N[(t$95$4 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(t$95$2 + N[(N[(0.5 * N[(t$95$4 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00000000000000008e-5

   1. Initial program 73.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 82.1

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 82.1%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 93.2

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 93.2%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. /-lowering-/.f64 50.5

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 50.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000008e-5 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00005000000000011

   1. Initial program 95.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 28.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      12. sqrt-lowering-sqrt.f64 32.2

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 32.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   if 1.00005000000000011 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 93.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. associate--l+ N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-inverses N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{1}}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{1}{\color{blue}{\sqrt{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{1}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 93.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 93.9%

      \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.00005:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(t\_4 + \left(t\_1 - \sqrt{y}\right)\right) \leq 10^{-5}:\\ \;\;\;\;t\_3 + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(t\_4 + \frac{1}{\sqrt{y} + t\_1}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t\_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= (+ (- t_2 (sqrt z)) (+ t_4 (- t_1 (sqrt y)))) 1e-5)
     (+ t_3 (* 0.5 (+ (sqrt (/ 1.0 z)) (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y))))))
     (+
      t_3
      (+
       (+ t_4 (/ 1.0 (+ (sqrt y) t_1)))
       (/ (- (+ 1.0 z) z) (+ (sqrt z) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (((t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)))) <= 1e-5) {
		tmp = t_3 + (0.5 * (sqrt((1.0 / z)) + (sqrt((1.0 / x)) + sqrt((1.0 / y)))));
	} else {
		tmp = t_3 + ((t_4 + (1.0 / (sqrt(y) + t_1))) + (((1.0 + z) - z) / (sqrt(z) + t_2)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = sqrt((1.0d0 + x)) - sqrt(x)
    if (((t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)))) <= 1d-5) then
        tmp = t_3 + (0.5d0 * (sqrt((1.0d0 / z)) + (sqrt((1.0d0 / x)) + sqrt((1.0d0 / y)))))
    else
        tmp = t_3 + ((t_4 + (1.0d0 / (sqrt(y) + t_1))) + (((1.0d0 + z) - z) / (sqrt(z) + t_2)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (((t_2 - Math.sqrt(z)) + (t_4 + (t_1 - Math.sqrt(y)))) <= 1e-5) {
		tmp = t_3 + (0.5 * (Math.sqrt((1.0 / z)) + (Math.sqrt((1.0 / x)) + Math.sqrt((1.0 / y)))));
	} else {
		tmp = t_3 + ((t_4 + (1.0 / (Math.sqrt(y) + t_1))) + (((1.0 + z) - z) / (Math.sqrt(z) + t_2)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if ((t_2 - math.sqrt(z)) + (t_4 + (t_1 - math.sqrt(y)))) <= 1e-5:
		tmp = t_3 + (0.5 * (math.sqrt((1.0 / z)) + (math.sqrt((1.0 / x)) + math.sqrt((1.0 / y)))))
	else:
		tmp = t_3 + ((t_4 + (1.0 / (math.sqrt(y) + t_1))) + (((1.0 + z) - z) / (math.sqrt(z) + t_2)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(z)) + Float64(t_4 + Float64(t_1 - sqrt(y)))) <= 1e-5)
		tmp = Float64(t_3 + Float64(0.5 * Float64(sqrt(Float64(1.0 / z)) + Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y))))));
	else
		tmp = Float64(t_3 + Float64(Float64(t_4 + Float64(1.0 / Float64(sqrt(y) + t_1))) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_2))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (((t_2 - sqrt(z)) + (t_4 + (t_1 - sqrt(y)))) <= 1e-5)
		tmp = t_3 + (0.5 * (sqrt((1.0 / z)) + (sqrt((1.0 / x)) + sqrt((1.0 / y)))));
	else
		tmp = t_3 + ((t_4 + (1.0 / (sqrt(y) + t_1))) + (((1.0 + z) - z) / (sqrt(z) + t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-5], N[(t$95$3 + N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(t$95$4 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00000000000000008e-5

   1. Initial program 50.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 66.7

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 66.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 87.6

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 87.6%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. /-lowering-/.f64 93.6

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 93.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000008e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{\color{blue}{1 + y}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{1 + y} \cdot \sqrt{\color{blue}{1 + y}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. associate--l+ N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-inverses N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1}}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 98.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 10^{-5}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := z + \left(1 + z\right)\\ t_6 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 + \left(\left(t\_6 - \sqrt{x}\right) + t\_1\right) \leq 0.999996:\\ \;\;\;\;t\_4 + \left(t\_3 + \frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + t\_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(t\_1 + \left(1 - \sqrt{x}\right)\right) + \frac{t\_5}{t\_5 \cdot \left(\sqrt{z} + t\_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (+ z (+ 1.0 z)))
        (t_6 (sqrt (+ 1.0 x))))
   (if (<= (+ t_3 (+ (- t_6 (sqrt x)) t_1)) 0.999996)
     (+
      t_4
      (+
       t_3
       (/
        (+ (sqrt y) (+ (sqrt x) 2.0))
        (* (+ 1.0 (sqrt y)) (+ (sqrt x) t_6)))))
     (+ t_4 (+ (+ t_1 (- 1.0 (sqrt x))) (/ t_5 (* t_5 (+ (sqrt z) t_2))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = z + (1.0 + z);
	double t_6 = sqrt((1.0 + x));
	double tmp;
	if ((t_3 + ((t_6 - sqrt(x)) + t_1)) <= 0.999996) {
		tmp = t_4 + (t_3 + ((sqrt(y) + (sqrt(x) + 2.0)) / ((1.0 + sqrt(y)) * (sqrt(x) + t_6))));
	} else {
		tmp = t_4 + ((t_1 + (1.0 - sqrt(x))) + (t_5 / (t_5 * (sqrt(z) + t_2))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y)) - sqrt(y)
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = z + (1.0d0 + z)
    t_6 = sqrt((1.0d0 + x))
    if ((t_3 + ((t_6 - sqrt(x)) + t_1)) <= 0.999996d0) then
        tmp = t_4 + (t_3 + ((sqrt(y) + (sqrt(x) + 2.0d0)) / ((1.0d0 + sqrt(y)) * (sqrt(x) + t_6))))
    else
        tmp = t_4 + ((t_1 + (1.0d0 - sqrt(x))) + (t_5 / (t_5 * (sqrt(z) + t_2))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = z + (1.0 + z);
	double t_6 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_3 + ((t_6 - Math.sqrt(x)) + t_1)) <= 0.999996) {
		tmp = t_4 + (t_3 + ((Math.sqrt(y) + (Math.sqrt(x) + 2.0)) / ((1.0 + Math.sqrt(y)) * (Math.sqrt(x) + t_6))));
	} else {
		tmp = t_4 + ((t_1 + (1.0 - Math.sqrt(x))) + (t_5 / (t_5 * (Math.sqrt(z) + t_2))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = z + (1.0 + z)
	t_6 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_3 + ((t_6 - math.sqrt(x)) + t_1)) <= 0.999996:
		tmp = t_4 + (t_3 + ((math.sqrt(y) + (math.sqrt(x) + 2.0)) / ((1.0 + math.sqrt(y)) * (math.sqrt(x) + t_6))))
	else:
		tmp = t_4 + ((t_1 + (1.0 - math.sqrt(x))) + (t_5 / (t_5 * (math.sqrt(z) + t_2))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = Float64(z + Float64(1.0 + z))
	t_6 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_3 + Float64(Float64(t_6 - sqrt(x)) + t_1)) <= 0.999996)
		tmp = Float64(t_4 + Float64(t_3 + Float64(Float64(sqrt(y) + Float64(sqrt(x) + 2.0)) / Float64(Float64(1.0 + sqrt(y)) * Float64(sqrt(x) + t_6)))));
	else
		tmp = Float64(t_4 + Float64(Float64(t_1 + Float64(1.0 - sqrt(x))) + Float64(t_5 / Float64(t_5 * Float64(sqrt(z) + t_2)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y)) - sqrt(y);
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = z + (1.0 + z);
	t_6 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((t_3 + ((t_6 - sqrt(x)) + t_1)) <= 0.999996)
		tmp = t_4 + (t_3 + ((sqrt(y) + (sqrt(x) + 2.0)) / ((1.0 + sqrt(y)) * (sqrt(x) + t_6))));
	else
		tmp = t_4 + ((t_1 + (1.0 - sqrt(x))) + (t_5 / (t_5 * (sqrt(z) + t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(z + N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(N[(t$95$6 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 0.999996], N[(t$95$4 + N[(t$95$3 + N[(N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(t$95$1 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / N[(t$95$5 * N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999995999999999996

   1. Initial program 58.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 62.4%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 70.9

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 70.9%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\frac{2 + \left(\sqrt{x} + \sqrt{y}\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{2 + \left(\sqrt{x} + \sqrt{y}\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(2 + \sqrt{x}\right) + \sqrt{y}}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(2 + \sqrt{x}\right) + \sqrt{y}}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(2 + \sqrt{x}\right)} + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \color{blue}{\sqrt{x}}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \color{blue}{\sqrt{y}}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\color{blue}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\color{blue}{\left(1 + \sqrt{y}\right)} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \color{blue}{\sqrt{y}}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \color{blue}{\left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\color{blue}{\sqrt{x}} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \color{blue}{\sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 85.1

          \[\leadsto \left(\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{\color{blue}{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 85.1%

       \[\leadsto \left(\color{blue}{\frac{\left(2 + \sqrt{x}\right) + \sqrt{y}}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.999995999999999996 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 57.0

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 57.0%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(z + \left(1 + z\right)\right) \cdot 1}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(z + \left(1 + z\right)\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.999996:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{\sqrt{y} + \left(\sqrt{x} + 2\right)}{\left(1 + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{z + \left(1 + z\right)}{\left(z + \left(1 + z\right)\right) \cdot \left(\sqrt{z} + \sqrt{1 + z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\\ \mathbf{if}\;t\_2 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_2 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y)))))
   (if (<= t_2 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_2 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 1.0) (sqrt x))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (- (+ 1.0 t_1) (+ (sqrt x) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = (sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y));
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_2 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), 1.0) - sqrt(x);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((1.0 + t_1) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y)))
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_2 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), 1.0) - sqrt(x));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(1.0 + t_1) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 74.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 80.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 80.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 13.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 13.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00005000000000011

   1. Initial program 95.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 51.0

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.0%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 53.6

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 53.6%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 29.8

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 29.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 18.5

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 18.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 93.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 56.8

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 56.8%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\\ \mathbf{if}\;t\_2 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_2 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y)))))
   (if (<= t_2 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_2 1.00005)
       (- (fma 0.5 (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 1.0) (sqrt x))
       (- (+ 1.0 t_1) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = (sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y));
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_2 <= 1.00005) {
		tmp = fma(0.5, (sqrt((1.0 / z)) + sqrt((1.0 / y))), 1.0) - sqrt(x);
	} else {
		tmp = (1.0 + t_1) - (sqrt(x) + sqrt(y));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y)))
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_2 <= 1.00005)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))), 1.0) - sqrt(x));
	else
		tmp = Float64(Float64(1.0 + t_1) - Float64(sqrt(x) + sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.00005], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 74.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 80.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 80.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 13.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 13.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.00005000000000011

   1. Initial program 95.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 51.0

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.0%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 53.6

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 53.6%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. distribute-lft-out N/A

          \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. sqrt-lowering-sqrt.f64 29.8

           \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 29.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + 1\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, 1\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 18.5

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \color{blue}{\sqrt{x}} \]

   14. Simplified 18.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, 1\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 93.9

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 56.8

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 56.8%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       7. sqrt-lowering-sqrt.f64 40.1

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   11. Simplified 40.1%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{y} + t\_1\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := z + \left(1 + z\right)\\ \mathbf{if}\;y \leq 7400:\\ \;\;\;\;t\_3 + \left(\left(\left(t\_1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{t\_4}{t\_4 \cdot \left(\sqrt{z} + \sqrt{1 + z}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\frac{\left(1 + \sqrt{x}\right) + t\_2}{t\_2 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z} - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (sqrt y) t_1))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (+ z (+ 1.0 z))))
   (if (<= y 7400.0)
     (+
      t_3
      (+
       (+ (- t_1 (sqrt y)) (- 1.0 (sqrt x)))
       (/ t_4 (* t_4 (+ (sqrt z) (sqrt (+ 1.0 z)))))))
     (+
      t_3
      (+
       (/ (+ (+ 1.0 (sqrt x)) t_2) (* t_2 (+ (sqrt x) (sqrt (+ 1.0 x)))))
       (- (sqrt z) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt(y) + t_1;
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = z + (1.0 + z);
	double tmp;
	if (y <= 7400.0) {
		tmp = t_3 + (((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (t_4 / (t_4 * (sqrt(z) + sqrt((1.0 + z))))));
	} else {
		tmp = t_3 + ((((1.0 + sqrt(x)) + t_2) / (t_2 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt(z) - sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt(y) + t_1
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    t_4 = z + (1.0d0 + z)
    if (y <= 7400.0d0) then
        tmp = t_3 + (((t_1 - sqrt(y)) + (1.0d0 - sqrt(x))) + (t_4 / (t_4 * (sqrt(z) + sqrt((1.0d0 + z))))))
    else
        tmp = t_3 + ((((1.0d0 + sqrt(x)) + t_2) / (t_2 * (sqrt(x) + sqrt((1.0d0 + x))))) + (sqrt(z) - sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt(y) + t_1;
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_4 = z + (1.0 + z);
	double tmp;
	if (y <= 7400.0) {
		tmp = t_3 + (((t_1 - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + (t_4 / (t_4 * (Math.sqrt(z) + Math.sqrt((1.0 + z))))));
	} else {
		tmp = t_3 + ((((1.0 + Math.sqrt(x)) + t_2) / (t_2 * (Math.sqrt(x) + Math.sqrt((1.0 + x))))) + (Math.sqrt(z) - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt(y) + t_1
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_4 = z + (1.0 + z)
	tmp = 0
	if y <= 7400.0:
		tmp = t_3 + (((t_1 - math.sqrt(y)) + (1.0 - math.sqrt(x))) + (t_4 / (t_4 * (math.sqrt(z) + math.sqrt((1.0 + z))))))
	else:
		tmp = t_3 + ((((1.0 + math.sqrt(x)) + t_2) / (t_2 * (math.sqrt(x) + math.sqrt((1.0 + x))))) + (math.sqrt(z) - math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(y) + t_1)
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = Float64(z + Float64(1.0 + z))
	tmp = 0.0
	if (y <= 7400.0)
		tmp = Float64(t_3 + Float64(Float64(Float64(t_1 - sqrt(y)) + Float64(1.0 - sqrt(x))) + Float64(t_4 / Float64(t_4 * Float64(sqrt(z) + sqrt(Float64(1.0 + z)))))));
	else
		tmp = Float64(t_3 + Float64(Float64(Float64(Float64(1.0 + sqrt(x)) + t_2) / Float64(t_2 * Float64(sqrt(x) + sqrt(Float64(1.0 + x))))) + Float64(sqrt(z) - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt(y) + t_1;
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	t_4 = z + (1.0 + z);
	tmp = 0.0;
	if (y <= 7400.0)
		tmp = t_3 + (((t_1 - sqrt(y)) + (1.0 - sqrt(x))) + (t_4 / (t_4 * (sqrt(z) + sqrt((1.0 + z))))));
	else
		tmp = t_3 + ((((1.0 + sqrt(x)) + t_2) / (t_2 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt(z) - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z + N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7400.0], N[(t$95$3 + N[(N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / N[(t$95$4 * N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(N[(N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(t$95$2 * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 7400

   1. Initial program 97.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 48.8

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 48.8%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \frac{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(1 + z\right) + z}}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(1 + z\right) \cdot \left(1 + z\right) - z \cdot z}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(\left(1 + z\right) + z\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 48.8%

      \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \color{blue}{\frac{\left(z + \left(1 + z\right)\right) \cdot 1}{\left(\sqrt{1 + z} + \sqrt{z}\right) \cdot \left(z + \left(1 + z\right)\right)}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 7400 < y

   1. Initial program 85.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 87.3%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 90.4

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 90.4%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\color{blue}{\sqrt{z}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 48.6

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\color{blue}{\sqrt{z}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 48.6%

       \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\color{blue}{\sqrt{z}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7400:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \frac{z + \left(1 + z\right)}{\left(z + \left(1 + z\right)\right) \cdot \left(\sqrt{z} + \sqrt{1 + z}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{1 + y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z} - \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{1 + y}\\ \left(\frac{\left(1 + \sqrt{x}\right) + t\_1}{t\_1 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt (+ 1.0 y)))))
   (+
    (+
     (/ (+ (+ 1.0 (sqrt x)) t_1) (* t_1 (+ (sqrt x) (sqrt (+ 1.0 x)))))
     (- (sqrt (+ 1.0 z)) (sqrt z)))
    (- (sqrt (+ 1.0 t)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt((1.0 + y));
	return ((((1.0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = sqrt(y) + sqrt((1.0d0 + y))
    code = ((((1.0d0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0d0 + x))))) + (sqrt((1.0d0 + z)) - sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(y) + Math.sqrt((1.0 + y));
	return ((((1.0 + Math.sqrt(x)) + t_1) / (t_1 * (Math.sqrt(x) + Math.sqrt((1.0 + x))))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt((1.0 + y))
	return ((((1.0 + math.sqrt(x)) + t_1) / (t_1 * (math.sqrt(x) + math.sqrt((1.0 + x))))) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(Float64(1.0 + y)))
	return Float64(Float64(Float64(Float64(Float64(1.0 + sqrt(x)) + t_1) / Float64(t_1 * Float64(sqrt(x) + sqrt(Float64(1.0 + x))))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt((1.0 + y));
	tmp = ((((1.0 + sqrt(x)) + t_1) / (t_1 * (sqrt(x) + sqrt((1.0 + x))))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(t$95$1 * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 91.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. flip-- N/A

      \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. frac-add N/A

      \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied egg-rr 92.3%

   \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. +-lowering-+.f64 75.3

      \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Simplified 75.3%

   \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Final simplification 75.3%

   \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{1 + y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
9. Add Preprocessing

Alternative 17: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\\ \mathbf{if}\;t\_2 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y)))))
   (if (<= t_2 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_2 1.0)
       (- (- (sqrt (+ 1.0 t)) (sqrt t)) (+ (sqrt x) -1.0))
       (- (+ 1.0 t_1) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = (sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y));
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_2 <= 1.0) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) - (sqrt(x) + -1.0);
	} else {
		tmp = (1.0 + t_1) - (sqrt(x) + sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = (sqrt((1.0d0 + x)) - sqrt(x)) + (t_1 - sqrt(y))
    if (t_2 <= 0.1d0) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else if (t_2 <= 1.0d0) then
        tmp = (sqrt((1.0d0 + t)) - sqrt(t)) - (sqrt(x) + (-1.0d0))
    else
        tmp = (1.0d0 + t_1) - (sqrt(x) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = (Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (t_1 - Math.sqrt(y));
	double tmp;
	if (t_2 <= 0.1) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else if (t_2 <= 1.0) {
		tmp = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) - (Math.sqrt(x) + -1.0);
	} else {
		tmp = (1.0 + t_1) - (Math.sqrt(x) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = (math.sqrt((1.0 + x)) - math.sqrt(x)) + (t_1 - math.sqrt(y))
	tmp = 0
	if t_2 <= 0.1:
		tmp = 0.5 * math.sqrt((1.0 / x))
	elif t_2 <= 1.0:
		tmp = (math.sqrt((1.0 + t)) - math.sqrt(t)) - (math.sqrt(x) + -1.0)
	else:
		tmp = (1.0 + t_1) - (math.sqrt(x) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y)))
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_2 <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - Float64(sqrt(x) + -1.0));
	else
		tmp = Float64(Float64(1.0 + t_1) - Float64(sqrt(x) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = (sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y));
	tmp = 0.0;
	if (t_2 <= 0.1)
		tmp = 0.5 * sqrt((1.0 / x));
	elseif (t_2 <= 1.0)
		tmp = (sqrt((1.0 + t)) - sqrt(t)) - (sqrt(x) + -1.0);
	else
		tmp = (1.0 + t_1) - (sqrt(x) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 74.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 80.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 80.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 13.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 13.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 50.8

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 50.8%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 9.0

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 9.0%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. sqrt-lowering-sqrt.f64 29.2

          \[\leadsto \left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 29.2%

       \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 91.7

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 91.7%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 55.0

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 55.0%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       7. sqrt-lowering-sqrt.f64 38.5

          \[\leadsto \left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   11. Simplified 38.5%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_1 \leq 1.5:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))))
   (if (<= t_1 0.1)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_1 1.5)
       (- (- (sqrt (+ 1.0 t)) (sqrt t)) (+ (sqrt x) -1.0))
       (- (fma y 0.5 2.0) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (sqrt((1.0 + x)) - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y));
	double tmp;
	if (t_1 <= 0.1) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_1 <= 1.5) {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) - (sqrt(x) + -1.0);
	} else {
		tmp = fma(y, 0.5, 2.0) - (sqrt(x) + sqrt(y));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_1 <= 1.5)
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - Float64(sqrt(x) + -1.0));
	else
		tmp = Float64(fma(y, 0.5, 2.0) - Float64(sqrt(x) + sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.1], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.10000000000000001

   1. Initial program 74.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 80.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 80.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 13.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 13.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.10000000000000001 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.5

   1. Initial program 95.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 50.8

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 50.8%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 9.5

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 9.5%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. sqrt-lowering-sqrt.f64 28.5

          \[\leadsto \left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 28.5%

       \[\leadsto \color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.5 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 98.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.2

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 97.2%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 59.9

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 59.9

          \[\leadsto \left(\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 59.9%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       7. sqrt-lowering-sqrt.f64 42.0

          \[\leadsto \mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   14. Simplified 42.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.1:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.5:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.1:\\ \;\;\;\;t\_1 + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.1)
     (+ t_1 (* 0.5 (+ (sqrt (/ 1.0 z)) (+ (sqrt (/ 1.0 x)) (sqrt (/ 1.0 y))))))
     (+
      t_1
      (+
       (- (sqrt (+ 1.0 z)) (sqrt z))
       (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (fma x 0.5 (- 1.0 (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.1) {
		tmp = t_1 + (0.5 * (sqrt((1.0 / z)) + (sqrt((1.0 / x)) + sqrt((1.0 / y)))));
	} else {
		tmp = t_1 + ((sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((1.0 + y)) - sqrt(y)) + fma(x, 0.5, (1.0 - sqrt(x)))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.1)
		tmp = Float64(t_1 + Float64(0.5 * Float64(sqrt(Float64(1.0 / z)) + Float64(sqrt(Float64(1.0 / x)) + sqrt(Float64(1.0 / y))))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + fma(x, 0.5, Float64(1.0 - sqrt(x))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.1], N[(t$95$1 + N[(0.5 * N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.10000000000000001

   1. Initial program 84.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 88.5

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 88.5%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 49.6

         \[\leadsto \left(0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 49.6%

      \[\leadsto \left(\color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{z}}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \color{blue}{\left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\color{blue}{\sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{\frac{1}{x}}} + \sqrt{\frac{1}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. /-lowering-/.f64 28.6

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\color{blue}{\frac{1}{y}}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 28.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 96.1

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.1%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 0.5 \cdot \left(\sqrt{\frac{1}{z}} + \left(\sqrt{\frac{1}{x}} + \sqrt{\frac{1}{y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 - \sqrt{x} \leq 0.999996:\\ \;\;\;\;t\_3 + \left(t\_1 + \frac{1}{\sqrt{x} + t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_2 (sqrt x)) 0.999996)
     (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt x) t_2))))
     (+ t_3 (+ t_1 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.999996) {
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	} else {
		tmp = t_3 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_2 - sqrt(x)) <= 0.999996d0) then
        tmp = t_3 + (t_1 + (1.0d0 / (sqrt(x) + t_2)))
    else
        tmp = t_3 + (t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) + (1.0d0 - sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.999996) {
		tmp = t_3 + (t_1 + (1.0 / (Math.sqrt(x) + t_2)));
	} else {
		tmp = t_3 + (t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (1.0 - Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.999996:
		tmp = t_3 + (t_1 + (1.0 / (math.sqrt(x) + t_2)))
	else:
		tmp = t_3 + (t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.999996)
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + t_2))));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(1.0 - sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_2 - sqrt(x)) <= 0.999996)
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	else
		tmp = t_3 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.999996], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.999995999999999996

   1. Initial program 85.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right) + \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y + 1} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 87.0%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(x + 1\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\left(1 + y\right) - y\right)\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 52.7

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{1 + y}}\right)}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 52.7%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 49.5

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 49.5%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.999995999999999996 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.5

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.5%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.999996:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 47.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.35e+39)
   (- (fma y 0.5 2.0) (+ (sqrt x) (sqrt y)))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.35e+39) {
		tmp = fma(y, 0.5, 2.0) - (sqrt(x) + sqrt(y));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.35e+39)
		tmp = Float64(fma(y, 0.5, 2.0) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.35e+39], N[(N[(y * 0.5 + 2.0), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000002e39

   1. Initial program 93.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 49.1

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 49.1%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 29.7

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 29.7%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 26.6

          \[\leadsto \left(\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 26.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   13. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot y + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\color{blue}{y \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) \]

       7. sqrt-lowering-sqrt.f64 19.2

          \[\leadsto \mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   14. Simplified 19.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 1.35000000000000002e39 < y

   1. Initial program 87.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 44.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 44.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 10.7

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 10.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 12.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 4.6e-43) (* 0.5 (sqrt (/ 1.0 y))) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.6e-43) {
		tmp = 0.5 * sqrt((1.0 / y));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 4.6d-43) then
        tmp = 0.5d0 * sqrt((1.0d0 / y))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.6e-43) {
		tmp = 0.5 * Math.sqrt((1.0 / y));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 4.6e-43:
		tmp = 0.5 * math.sqrt((1.0 / y))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4.6e-43)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / y)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 4.6e-43)
		tmp = 0.5 * sqrt((1.0 / y));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 4.6e-43], N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999998e-43

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.8%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 55.6

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 55.6%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{y}}} \]

       3. /-lowering-/.f64 6.9

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{y}}} \]

   11. Simplified 6.9%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{y}}} \]

   if 4.5999999999999998e-43 < x

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 80.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 80.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. /-lowering-/.f64 10.4

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 10.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{x}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* 0.5 (sqrt (/ 1.0 x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / x));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 * sqrt((1.0d0 / x))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.sqrt((1.0 / x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate--l+ N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. sqrt-lowering-sqrt.f64 46.4

      \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 46.4%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   3. /-lowering-/.f64 8.5

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

8. Simplified 8.5%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
9. Add Preprocessing

Alternative 24: 1.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0 - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 0.0 (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.0 - sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.0 - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.0 - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.0 - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.0 - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. associate--r+ N/A

      \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. associate-+l- N/A

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. sqrt-lowering-sqrt.f64 49.5

       \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 49.5%

   \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \sqrt{x}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \sqrt{x}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \sqrt{x}} \]

   4. sqrt-lowering-sqrt.f64 1.6

      \[\leadsto 0 - \color{blue}{\sqrt{x}} \]

8. Simplified 1.6%

   \[\leadsto \color{blue}{0 - \sqrt{x}} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)} \]

   3. sqrt-lowering-sqrt.f64 1.6

      \[\leadsto -\color{blue}{\sqrt{x}} \]

10. Applied egg-rr 1.6%

    \[\leadsto \color{blue}{-\sqrt{x}} \]

11. Final simplification 1.6%

    \[\leadsto 0 - \sqrt{x} \]
12. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))