Main:z from

Percentage Accurate: 91.7% → 97.7%

Time: 26.3s

Alternatives: 23

Speedup: 0.5×

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: 91.7% 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: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := \sqrt{1 + z}\\ t_6 := t\_5 - \sqrt{z}\\ t_7 := t\_1 - \sqrt{x}\\ t_8 := t\_4 + \left(\left(t\_7 + \left(t\_2 - \sqrt{y}\right)\right) + t\_6\right)\\ \mathbf{if}\;t\_8 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_8 \leq 1.002:\\ \;\;\;\;t\_4 + \left(t\_6 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_7\right)\right)\\ \mathbf{elif}\;t\_8 \leq 3:\\ \;\;\;\;\left(\left(1 + t\_2\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + t\_5}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + t\_5\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\right) + t\_3\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6 (- t_5 (sqrt z)))
        (t_7 (- t_1 (sqrt x)))
        (t_8 (+ t_4 (+ (+ t_7 (- t_2 (sqrt y))) t_6))))
   (if (<= t_8 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_8 1.002)
       (+ t_4 (+ t_6 (fma 0.5 (sqrt (/ 1.0 y)) t_7)))
       (if (<= t_8 3.0)
         (-
          (+ (+ 1.0 t_2) (fma x 0.5 (/ 1.0 (+ (sqrt z) t_5))))
          (+ (sqrt x) (sqrt y)))
         (+
          (+
           (+ t_1 t_5)
           (- 1.0 (+ (sqrt x) (+ (sqrt z) (+ (sqrt y) (sqrt t))))))
          t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t));
	double t_4 = t_3 - sqrt(t);
	double t_5 = sqrt((1.0 + z));
	double t_6 = t_5 - sqrt(z);
	double t_7 = t_1 - sqrt(x);
	double t_8 = t_4 + ((t_7 + (t_2 - sqrt(y))) + t_6);
	double tmp;
	if (t_8 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_8 <= 1.002) {
		tmp = t_4 + (t_6 + fma(0.5, sqrt((1.0 / y)), t_7));
	} else if (t_8 <= 3.0) {
		tmp = ((1.0 + t_2) + fma(x, 0.5, (1.0 / (sqrt(z) + t_5)))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((t_1 + t_5) + (1.0 - (sqrt(x) + (sqrt(z) + (sqrt(y) + sqrt(t)))))) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(t_5 - sqrt(z))
	t_7 = Float64(t_1 - sqrt(x))
	t_8 = Float64(t_4 + Float64(Float64(t_7 + Float64(t_2 - sqrt(y))) + t_6))
	tmp = 0.0
	if (t_8 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_8 <= 1.002)
		tmp = Float64(t_4 + Float64(t_6 + fma(0.5, sqrt(Float64(1.0 / y)), t_7)));
	elseif (t_8 <= 3.0)
		tmp = Float64(Float64(Float64(1.0 + t_2) + fma(x, 0.5, Float64(1.0 / Float64(sqrt(z) + t_5)))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(t_1 + t_5) + Float64(1.0 - Float64(sqrt(x) + Float64(sqrt(z) + Float64(sqrt(y) + sqrt(t)))))) + t_3);
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$4 + N[(N[(t$95$7 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$8, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$8, 1.002], N[(t$95$4 + N[(t$95$6 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$8, 3.0], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(x * 0.5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + t$95$5), $MachinePrecision] + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 69.1

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

   5. Simplified 69.1%

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

   if 1.002 < (+.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))) < 3

   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. 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 97.6

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

   4. Applied egg-rr 97.6%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 28.0

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

   7. Simplified 28.0%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 24.1%

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

   if 3 < (+.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 97.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

   5. Simplified 90.7%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

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

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

   7. Applied egg-rr 90.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{x + 1} - \sqrt{x}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 3:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \sqrt{1 + z}\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\right) + \sqrt{1 + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + z}\\ t_5 := \frac{1}{\sqrt{z} + t\_4}\\ t_6 := t\_1 - \sqrt{x}\\ t_7 := \left(t\_3 - \sqrt{t}\right) + \left(\left(t\_6 + \left(t\_2 - \sqrt{y}\right)\right) + \left(t\_4 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_7 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_7 \leq 1.002:\\ \;\;\;\;t\_6 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_5\right)\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(1 + t\_2\right) + \mathsf{fma}\left(x, 0.5, t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + t\_4\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\right) + t\_3\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (/ 1.0 (+ (sqrt z) t_4)))
        (t_6 (- t_1 (sqrt x)))
        (t_7
         (+ (- t_3 (sqrt t)) (+ (+ t_6 (- t_2 (sqrt y))) (- t_4 (sqrt z))))))
   (if (<= t_7 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_7 1.002)
       (+ t_6 (fma 0.5 (sqrt (/ 1.0 y)) t_5))
       (if (<= t_7 3.0)
         (- (+ (+ 1.0 t_2) (fma x 0.5 t_5)) (+ (sqrt x) (sqrt y)))
         (+
          (+
           (+ t_1 t_4)
           (- 1.0 (+ (sqrt x) (+ (sqrt z) (+ (sqrt y) (sqrt t))))))
          t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((1.0 + z));
	double t_5 = 1.0 / (sqrt(z) + t_4);
	double t_6 = t_1 - sqrt(x);
	double t_7 = (t_3 - sqrt(t)) + ((t_6 + (t_2 - sqrt(y))) + (t_4 - sqrt(z)));
	double tmp;
	if (t_7 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_7 <= 1.002) {
		tmp = t_6 + fma(0.5, sqrt((1.0 / y)), t_5);
	} else if (t_7 <= 3.0) {
		tmp = ((1.0 + t_2) + fma(x, 0.5, t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((t_1 + t_4) + (1.0 - (sqrt(x) + (sqrt(z) + (sqrt(y) + sqrt(t)))))) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(1.0 / Float64(sqrt(z) + t_4))
	t_6 = Float64(t_1 - sqrt(x))
	t_7 = Float64(Float64(t_3 - sqrt(t)) + Float64(Float64(t_6 + Float64(t_2 - sqrt(y))) + Float64(t_4 - sqrt(z))))
	tmp = 0.0
	if (t_7 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_7 <= 1.002)
		tmp = Float64(t_6 + fma(0.5, sqrt(Float64(1.0 / y)), t_5));
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(1.0 + t_2) + fma(x, 0.5, t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(t_1 + t_4) + Float64(1.0 - Float64(sqrt(x) + Float64(sqrt(z) + Float64(sqrt(y) + sqrt(t)))))) + t_3);
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$6 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.002], N[(t$95$6 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(x * 0.5 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + t$95$4), $MachinePrecision] + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. 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 93.2

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

   4. Applied egg-rr 93.2%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 29.6

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

   7. Simplified 29.6%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sqrt-lowering-sqrt.f64 57.7

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

   10. Simplified 57.7%

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

   if 1.002 < (+.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))) < 3

   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. 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 97.6

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

   4. Applied egg-rr 97.6%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 28.0

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

   7. Simplified 28.0%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 24.1%

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

   if 3 < (+.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 97.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

   5. Simplified 90.7%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

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

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

   7. Applied egg-rr 90.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 3:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \sqrt{1 + z}\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\right) + \sqrt{1 + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + z}\\ t_4 := \frac{1}{\sqrt{z} + t\_3}\\ t_5 := \sqrt{x + 1} - \sqrt{x}\\ t_6 := t\_2 + \left(\left(t\_5 + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 1.002:\\ \;\;\;\;t\_5 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_4\right)\\ \mathbf{elif}\;t\_6 \leq 2.99:\\ \;\;\;\;\left(\left(1 + t\_1\right) + \mathsf{fma}\left(x, 0.5, t\_4\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(1 + \left(t\_1 + \left(t\_3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (/ 1.0 (+ (sqrt z) t_3)))
        (t_5 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_6 (+ t_2 (+ (+ t_5 (- t_1 (sqrt y))) (- t_3 (sqrt z))))))
   (if (<= t_6 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 1.002)
       (+ t_5 (fma 0.5 (sqrt (/ 1.0 y)) t_4))
       (if (<= t_6 2.99)
         (- (+ (+ 1.0 t_1) (fma x 0.5 t_4)) (+ (sqrt x) (sqrt y)))
         (+
          t_2
          (+ 1.0 (+ t_1 (- t_3 (+ (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 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + z));
	double t_4 = 1.0 / (sqrt(z) + t_3);
	double t_5 = sqrt((x + 1.0)) - sqrt(x);
	double t_6 = t_2 + ((t_5 + (t_1 - sqrt(y))) + (t_3 - sqrt(z)));
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 1.002) {
		tmp = t_5 + fma(0.5, sqrt((1.0 / y)), t_4);
	} else if (t_6 <= 2.99) {
		tmp = ((1.0 + t_1) + fma(x, 0.5, t_4)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = t_2 + (1.0 + (t_1 + (t_3 - (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 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(1.0 / Float64(sqrt(z) + t_3))
	t_5 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_6 = Float64(t_2 + Float64(Float64(t_5 + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))))
	tmp = 0.0
	if (t_6 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 1.002)
		tmp = Float64(t_5 + fma(0.5, sqrt(Float64(1.0 / y)), t_4));
	elseif (t_6 <= 2.99)
		tmp = Float64(Float64(Float64(1.0 + t_1) + fma(x, 0.5, t_4)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(t_2 + Float64(1.0 + Float64(t_1 + Float64(t_3 - 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[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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 + N[(N[(t$95$5 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.002], N[(t$95$5 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.99], N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(x * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 + N[(t$95$1 + N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. 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 93.2

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

   4. Applied egg-rr 93.2%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 29.6

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

   7. Simplified 29.6%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sqrt-lowering-sqrt.f64 57.7

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

   10. Simplified 57.7%

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

   if 1.002 < (+.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))) < 2.9900000000000002

   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. 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 96.7

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

   4. Applied egg-rr 96.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 27.0

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

   7. Simplified 27.0%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 22.8%

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

   if 2.9900000000000002 < (+.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 98.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

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

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 69.1

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

   5. Simplified 69.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.99:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \frac{1}{\sqrt{z} + t\_3}\\ t_6 := \sqrt{x + 1} - \sqrt{x}\\ t_7 := t\_2 + \left(\left(t\_6 + \left(t\_1 - \sqrt{y}\right)\right) + t\_4\right)\\ \mathbf{if}\;t\_7 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_7 \leq 1.002:\\ \;\;\;\;t\_6 + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_5\right)\\ \mathbf{elif}\;t\_7 \leq 2.9995:\\ \;\;\;\;\left(\left(1 + t\_1\right) + \mathsf{fma}\left(x, 0.5, t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_4 + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (/ 1.0 (+ (sqrt z) t_3)))
        (t_6 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_7 (+ t_2 (+ (+ t_6 (- t_1 (sqrt y))) t_4))))
   (if (<= t_7 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_7 1.002)
       (+ t_6 (fma 0.5 (sqrt (/ 1.0 y)) t_5))
       (if (<= t_7 2.9995)
         (- (+ (+ 1.0 t_1) (fma x 0.5 t_5)) (+ (sqrt x) (sqrt y)))
         (+ t_2 (+ t_4 (- (fma x 0.5 2.0) (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 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = 1.0 / (sqrt(z) + t_3);
	double t_6 = sqrt((x + 1.0)) - sqrt(x);
	double t_7 = t_2 + ((t_6 + (t_1 - sqrt(y))) + t_4);
	double tmp;
	if (t_7 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_7 <= 1.002) {
		tmp = t_6 + fma(0.5, sqrt((1.0 / y)), t_5);
	} else if (t_7 <= 2.9995) {
		tmp = ((1.0 + t_1) + fma(x, 0.5, t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = t_2 + (t_4 + (fma(x, 0.5, 2.0) - 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(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = Float64(1.0 / Float64(sqrt(z) + t_3))
	t_6 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_7 = Float64(t_2 + Float64(Float64(t_6 + Float64(t_1 - sqrt(y))) + t_4))
	tmp = 0.0
	if (t_7 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_7 <= 1.002)
		tmp = Float64(t_6 + fma(0.5, sqrt(Float64(1.0 / y)), t_5));
	elseif (t_7 <= 2.9995)
		tmp = Float64(Float64(Float64(1.0 + t_1) + fma(x, 0.5, t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(t_2 + Float64(t_4 + Float64(fma(x, 0.5, 2.0) - 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[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[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 + N[(N[(t$95$6 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.002], N[(t$95$6 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.9995], N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(x * 0.5 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$4 + N[(N[(x * 0.5 + 2.0), $MachinePrecision] - N[Sqrt[y], $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. 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 93.2

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

   4. Applied egg-rr 93.2%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 29.6

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

   7. Simplified 29.6%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sqrt-lowering-sqrt.f64 57.7

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

   10. Simplified 57.7%

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

   if 1.002 < (+.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))) < 2.99949999999999983

   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. 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 96.7

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

   4. Applied egg-rr 96.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 26.9

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

   7. Simplified 26.9%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 22.8%

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

   if 2.99949999999999983 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 65.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 65.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

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

       1. sqrt-lowering-sqrt.f64 63.9

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 63.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.9995:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_4 - \sqrt{z}\\ t_6 := t\_3 + \left(\left(\left(t\_1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) + t\_5\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 1.002:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 2.9995:\\ \;\;\;\;\left(\left(1 + t\_2\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + t\_4}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_5 + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (- t_4 (sqrt z)))
        (t_6 (+ t_3 (+ (+ (- t_1 (sqrt x)) (- t_2 (sqrt y))) t_5))))
   (if (<= t_6 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 1.002)
       (+ t_3 (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x)))
       (if (<= t_6 2.9995)
         (-
          (+ (+ 1.0 t_2) (fma x 0.5 (/ 1.0 (+ (sqrt z) t_4))))
          (+ (sqrt x) (sqrt y)))
         (+ t_3 (+ t_5 (- (fma x 0.5 2.0) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_4 - sqrt(z);
	double t_6 = t_3 + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + t_5);
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 1.002) {
		tmp = t_3 + (fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x));
	} else if (t_6 <= 2.9995) {
		tmp = ((1.0 + t_2) + fma(x, 0.5, (1.0 / (sqrt(z) + t_4)))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = t_3 + (t_5 + (fma(x, 0.5, 2.0) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_4 - sqrt(z))
	t_6 = Float64(t_3 + Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y))) + t_5))
	tmp = 0.0
	if (t_6 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 1.002)
		tmp = Float64(t_3 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x)));
	elseif (t_6 <= 2.9995)
		tmp = Float64(Float64(Float64(1.0 + t_2) + fma(x, 0.5, Float64(1.0 / Float64(sqrt(z) + t_4)))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(t_3 + Float64(t_5 + Float64(fma(x, 0.5, 2.0) - 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[(x + 1.0), $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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.002], N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.9995], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(x * 0.5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$5 + N[(N[(x * 0.5 + 2.0), $MachinePrecision] - N[Sqrt[y], $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 17.2

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

   5. Simplified 17.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 39.6

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

   8. Simplified 39.6%

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

   if 1.002 < (+.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))) < 2.99949999999999983

   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. 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 96.7

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

   4. Applied egg-rr 96.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 26.9

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

   7. Simplified 26.9%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 22.8%

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

   if 2.99949999999999983 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 65.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 65.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

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

       1. sqrt-lowering-sqrt.f64 63.9

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 63.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.9995:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) + \mathsf{fma}\left(x, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_4 - \sqrt{z}\\ t_6 := t\_3 + \left(\left(\left(t\_1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) + t\_5\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 1.002:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 2.9995:\\ \;\;\;\;t\_1 + \left(\left(t\_2 + \frac{1}{\sqrt{z} + t\_4}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_5 + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (- t_4 (sqrt z)))
        (t_6 (+ t_3 (+ (+ (- t_1 (sqrt x)) (- t_2 (sqrt y))) t_5))))
   (if (<= t_6 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 1.002)
       (+ t_3 (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x)))
       (if (<= t_6 2.9995)
         (+ t_1 (- (+ t_2 (/ 1.0 (+ (sqrt z) t_4))) (sqrt y)))
         (+ t_3 (+ t_5 (- (fma x 0.5 2.0) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_4 - sqrt(z);
	double t_6 = t_3 + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + t_5);
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 1.002) {
		tmp = t_3 + (fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x));
	} else if (t_6 <= 2.9995) {
		tmp = t_1 + ((t_2 + (1.0 / (sqrt(z) + t_4))) - sqrt(y));
	} else {
		tmp = t_3 + (t_5 + (fma(x, 0.5, 2.0) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_4 - sqrt(z))
	t_6 = Float64(t_3 + Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y))) + t_5))
	tmp = 0.0
	if (t_6 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 1.002)
		tmp = Float64(t_3 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x)));
	elseif (t_6 <= 2.9995)
		tmp = Float64(t_1 + Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(z) + t_4))) - sqrt(y)));
	else
		tmp = Float64(t_3 + Float64(t_5 + Float64(fma(x, 0.5, 2.0) - 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[(x + 1.0), $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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.002], N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.9995], N[(t$95$1 + N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$5 + N[(N[(x * 0.5 + 2.0), $MachinePrecision] - N[Sqrt[y], $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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))) < 1.002

   1. Initial program 92.9%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 17.2

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

   5. Simplified 17.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 39.6

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

   8. Simplified 39.6%

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

   if 1.002 < (+.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))) < 2.99949999999999983

   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. 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 96.7

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

   4. Applied egg-rr 96.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 26.9

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

   7. Simplified 26.9%

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

   8. Taylor expanded in y around inf

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

      1. sqrt-lowering-sqrt.f64 28.1

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

   10. Simplified 28.1%

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

   if 2.99949999999999983 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 65.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 65.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

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

       1. sqrt-lowering-sqrt.f64 63.9

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 63.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.9995:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + z}\\ t_5 := t\_4 - \sqrt{z}\\ t_6 := t\_3 + \left(\left(\left(t\_1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right) + t\_5\right)\\ \mathbf{if}\;t\_6 \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 2.9995:\\ \;\;\;\;t\_1 + \left(\left(t\_2 + \frac{1}{\sqrt{z} + t\_4}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_5 + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (- t_4 (sqrt z)))
        (t_6 (+ t_3 (+ (+ (- t_1 (sqrt x)) (- t_2 (sqrt y))) t_5))))
   (if (<= t_6 0.2)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 1.002)
       (fma 0.5 (+ x (sqrt (/ 1.0 y))) (- 1.0 (sqrt x)))
       (if (<= t_6 2.9995)
         (+ t_1 (- (+ t_2 (/ 1.0 (+ (sqrt z) t_4))) (sqrt y)))
         (+ t_3 (+ t_5 (- (fma x 0.5 2.0) (sqrt y)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + z));
	double t_5 = t_4 - sqrt(z);
	double t_6 = t_3 + (((t_1 - sqrt(x)) + (t_2 - sqrt(y))) + t_5);
	double tmp;
	if (t_6 <= 0.2) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 1.002) {
		tmp = fma(0.5, (x + sqrt((1.0 / y))), (1.0 - sqrt(x)));
	} else if (t_6 <= 2.9995) {
		tmp = t_1 + ((t_2 + (1.0 / (sqrt(z) + t_4))) - sqrt(y));
	} else {
		tmp = t_3 + (t_5 + (fma(x, 0.5, 2.0) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(t_4 - sqrt(z))
	t_6 = Float64(t_3 + Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y))) + t_5))
	tmp = 0.0
	if (t_6 <= 0.2)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 1.002)
		tmp = fma(0.5, Float64(x + sqrt(Float64(1.0 / y))), Float64(1.0 - sqrt(x)));
	elseif (t_6 <= 2.9995)
		tmp = Float64(t_1 + Float64(Float64(t_2 + Float64(1.0 / Float64(sqrt(z) + t_4))) - sqrt(y)));
	else
		tmp = Float64(t_3 + Float64(t_5 + Float64(fma(x, 0.5, 2.0) - 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[(x + 1.0), $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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.2], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.002], N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.9995], N[(t$95$1 + N[(N[(t$95$2 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$5 + N[(N[(x * 0.5 + 2.0), $MachinePrecision] - N[Sqrt[y], $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))) < 0.20000000000000001

   1. Initial program 19.5%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 9.5%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 57.0

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

   11. Simplified 57.0%

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

   if 0.20000000000000001 < (+.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))) < 1.002

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 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 30.1

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

   5. Simplified 30.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 30.5

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

   8. Simplified 30.5%

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

   9. Taylor expanded in z around inf

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

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

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

       2. +-commutative N/A

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

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

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

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

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 30.5

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

   11. Simplified 30.5%

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

   12. Taylor expanded in t around inf

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

       8. sqrt-lowering-sqrt.f64 30.2

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

   14. Simplified 30.2%

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

   if 1.002 < (+.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))) < 2.99949999999999983

   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. 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 96.7

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

   4. Applied egg-rr 96.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 26.9

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

   7. Simplified 26.9%

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

   8. Taylor expanded in y around inf

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

      1. sqrt-lowering-sqrt.f64 28.1

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

   10. Simplified 28.1%

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

   if 2.99949999999999983 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 65.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 65.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

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

       1. sqrt-lowering-sqrt.f64 63.9

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 63.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.9995:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := t\_2 - \sqrt{y}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{1 + z}\\ t_6 := \left(t\_4 - \sqrt{t}\right) + \left(\left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + \left(t\_5 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_3 - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(t\_5 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, t\_4\right) + \left(3 - \left(\sqrt{z} + \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6
         (+ (- t_4 (sqrt t)) (+ (+ (- t_1 (sqrt x)) t_3) (- t_5 (sqrt z))))))
   (if (<= t_6 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 2.0)
       (+ t_1 (- t_3 (sqrt x)))
       (if (<= t_6 3.0)
         (- (+ t_5 (+ 1.0 t_2)) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
         (+
          (fma x 0.5 t_4)
          (- 3.0 (+ (sqrt z) (+ (sqrt t) (+ (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((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 - sqrt(y);
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt((1.0 + z));
	double t_6 = (t_4 - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_5 - sqrt(z)));
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 2.0) {
		tmp = t_1 + (t_3 - sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = fma(x, 0.5, t_4) + (3.0 - (sqrt(z) + (sqrt(t) + (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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(Float64(t_4 - sqrt(t)) + Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + Float64(t_5 - sqrt(z))))
	tmp = 0.0
	if (t_6 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 2.0)
		tmp = Float64(t_1 + Float64(t_3 - sqrt(x)));
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(t_5 + Float64(1.0 + t_2)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(fma(x, 0.5, t_4) + Float64(3.0 - Float64(sqrt(z) + Float64(sqrt(t) + 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(t$95$1 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(t$95$5 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5 + t$95$4), $MachinePrecision] + N[(3.0 - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 3.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. /-lowering-/.f64 35.3

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 63.6

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

   11. Simplified 63.6%

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

   if 4.9999999999999998e-8 < (+.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

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 23.0

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

   5. Simplified 23.0%

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

   6. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

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

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

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

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

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

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

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

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 25.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   if 2 < (+.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))) < 3

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 24.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 24.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 19.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 3 < (+.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 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 97.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 97.7%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 90.6

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 90.6%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) + 3\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) + \left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right) + \color{blue}{\left(3 - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

       10. associate-+r+ N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right) + \left(3 - \left(\sqrt{t} + \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) \]

       11. associate-+r+ N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right) + \left(3 - \color{blue}{\left(\left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) + \sqrt{z}\right)}\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right) + \left(3 - \color{blue}{\left(\left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right) + \sqrt{z}\right)}\right) \]

   11. Simplified 86.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) + \left(3 - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right) + \sqrt{z}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 3:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) + \left(3 - \left(\sqrt{z} + \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y} + \sqrt{z}\\ t_3 := \sqrt{1 + y}\\ t_4 := t\_3 - \sqrt{y}\\ t_5 := \sqrt{1 + t} - \sqrt{t}\\ t_6 := \sqrt{1 + z}\\ t_7 := t\_5 + \left(\left(\left(t\_1 - \sqrt{x}\right) + t\_4\right) + \left(t\_6 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_7 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_7 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_4 - \sqrt{x}\right)\\ \mathbf{elif}\;t\_7 \leq 2.99998:\\ \;\;\;\;\left(t\_6 + \left(1 + t\_3\right)\right) - \left(\sqrt{x} + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5 + \left(\mathsf{fma}\left(x, 0.5, 3 - \sqrt{x}\right) - t\_2\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (+ (sqrt y) (sqrt z)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (- t_3 (sqrt y)))
        (t_5 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_6 (sqrt (+ 1.0 z)))
        (t_7 (+ t_5 (+ (+ (- t_1 (sqrt x)) t_4) (- t_6 (sqrt z))))))
   (if (<= t_7 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_7 2.0)
       (+ t_1 (- t_4 (sqrt x)))
       (if (<= t_7 2.99998)
         (- (+ t_6 (+ 1.0 t_3)) (+ (sqrt x) t_2))
         (+ t_5 (- (fma x 0.5 (- 3.0 (sqrt x))) t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt(y) + sqrt(z);
	double t_3 = sqrt((1.0 + y));
	double t_4 = t_3 - sqrt(y);
	double t_5 = sqrt((1.0 + t)) - sqrt(t);
	double t_6 = sqrt((1.0 + z));
	double t_7 = t_5 + (((t_1 - sqrt(x)) + t_4) + (t_6 - sqrt(z)));
	double tmp;
	if (t_7 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_7 <= 2.0) {
		tmp = t_1 + (t_4 - sqrt(x));
	} else if (t_7 <= 2.99998) {
		tmp = (t_6 + (1.0 + t_3)) - (sqrt(x) + t_2);
	} else {
		tmp = t_5 + (fma(x, 0.5, (3.0 - sqrt(x))) - t_2);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(y) + sqrt(z))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(t_3 - sqrt(y))
	t_5 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_6 = sqrt(Float64(1.0 + z))
	t_7 = Float64(t_5 + Float64(Float64(Float64(t_1 - sqrt(x)) + t_4) + Float64(t_6 - sqrt(z))))
	tmp = 0.0
	if (t_7 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_7 <= 2.0)
		tmp = Float64(t_1 + Float64(t_4 - sqrt(x)));
	elseif (t_7 <= 2.99998)
		tmp = Float64(Float64(t_6 + Float64(1.0 + t_3)) - Float64(sqrt(x) + t_2));
	else
		tmp = Float64(t_5 + Float64(fma(x, 0.5, Float64(3.0 - sqrt(x))) - t_2));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(t$95$6 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.0], N[(t$95$1 + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.99998], N[(N[(t$95$6 + N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(N[(x * 0.5 + N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 3.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 35.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 63.6

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 23.0

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 23.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 25.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   if 2 < (+.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))) < 2.9999799999999999

   1. Initial program 83.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 27.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 20.5

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 20.5%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 2.9999799999999999 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 84.5

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 84.5%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 66.1

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 66.1%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\left(3 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(3 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(3 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 3\right)} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. associate--l+ N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(3 - \sqrt{x}\right)\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(3 - \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 3 - \sqrt{x}\right)} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{3 - \sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, 3 - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, 3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, 3 - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       11. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, 3 - \sqrt{x}\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       12. sqrt-lowering-sqrt.f64 41.5

           \[\leadsto \left(\mathsf{fma}\left(x, 0.5, 3 - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 41.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, 0.5, 3 - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2.99998:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(x, 0.5, 3 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := t\_2 - \sqrt{y}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{1 + z}\\ t_6 := \left(t\_4 - \sqrt{t}\right) + \left(\left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + \left(t\_5 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_3 - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(t\_5 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + t\_5\right) + t\_4\right) + \left(1 - \left(\sqrt{x} + \sqrt{t}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6
         (+ (- t_4 (sqrt t)) (+ (+ (- t_1 (sqrt x)) t_3) (- t_5 (sqrt z))))))
   (if (<= t_6 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_6 2.0)
       (+ t_1 (- t_3 (sqrt x)))
       (if (<= t_6 3.0)
         (- (+ t_5 (+ 1.0 t_2)) (+ (sqrt x) (+ (sqrt y) (sqrt z))))
         (+ (+ (+ t_1 t_5) t_4) (- 1.0 (+ (sqrt x) (sqrt t)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 - sqrt(y);
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt((1.0 + z));
	double t_6 = (t_4 - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_5 - sqrt(z)));
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_6 <= 2.0) {
		tmp = t_1 + (t_3 - sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = ((t_1 + t_5) + t_4) + (1.0 - (sqrt(x) + sqrt(t)));
	}
	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((x + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_2 - sqrt(y)
    t_4 = sqrt((1.0d0 + t))
    t_5 = sqrt((1.0d0 + z))
    t_6 = (t_4 - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_5 - sqrt(z)))
    if (t_6 <= 5d-8) then
        tmp = 0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / x)))
    else if (t_6 <= 2.0d0) then
        tmp = t_1 + (t_3 - sqrt(x))
    else if (t_6 <= 3.0d0) then
        tmp = (t_5 + (1.0d0 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = ((t_1 + t_5) + t_4) + (1.0d0 - (sqrt(x) + sqrt(t)))
    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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = t_2 - Math.sqrt(y);
	double t_4 = Math.sqrt((1.0 + t));
	double t_5 = Math.sqrt((1.0 + z));
	double t_6 = (t_4 - Math.sqrt(t)) + (((t_1 - Math.sqrt(x)) + t_3) + (t_5 - Math.sqrt(z)));
	double tmp;
	if (t_6 <= 5e-8) {
		tmp = 0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / x)));
	} else if (t_6 <= 2.0) {
		tmp = t_1 + (t_3 - Math.sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = (t_5 + (1.0 + t_2)) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = ((t_1 + t_5) + t_4) + (1.0 - (Math.sqrt(x) + Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((1.0 + y))
	t_3 = t_2 - math.sqrt(y)
	t_4 = math.sqrt((1.0 + t))
	t_5 = math.sqrt((1.0 + z))
	t_6 = (t_4 - math.sqrt(t)) + (((t_1 - math.sqrt(x)) + t_3) + (t_5 - math.sqrt(z)))
	tmp = 0
	if t_6 <= 5e-8:
		tmp = 0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / x)))
	elif t_6 <= 2.0:
		tmp = t_1 + (t_3 - math.sqrt(x))
	elif t_6 <= 3.0:
		tmp = (t_5 + (1.0 + t_2)) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = ((t_1 + t_5) + t_4) + (1.0 - (math.sqrt(x) + math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(Float64(t_4 - sqrt(t)) + Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + Float64(t_5 - sqrt(z))))
	tmp = 0.0
	if (t_6 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_6 <= 2.0)
		tmp = Float64(t_1 + Float64(t_3 - sqrt(x)));
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(t_5 + Float64(1.0 + t_2)) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(Float64(t_1 + t_5) + t_4) + Float64(1.0 - Float64(sqrt(x) + sqrt(t))));
	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((x + 1.0));
	t_2 = sqrt((1.0 + y));
	t_3 = t_2 - sqrt(y);
	t_4 = sqrt((1.0 + t));
	t_5 = sqrt((1.0 + z));
	t_6 = (t_4 - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_5 - sqrt(z)));
	tmp = 0.0;
	if (t_6 <= 5e-8)
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	elseif (t_6 <= 2.0)
		tmp = t_1 + (t_3 - sqrt(x));
	elseif (t_6 <= 3.0)
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = ((t_1 + t_5) + t_4) + (1.0 - (sqrt(x) + sqrt(t)));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(t$95$1 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(t$95$5 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + t$95$5), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[t], $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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 3.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 35.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 63.6

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 23.0

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 23.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 25.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   if 2 < (+.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))) < 3

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 24.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 24.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 19.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 3 < (+.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 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) + \left(\mathsf{neg}\left(\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) + 1\right)} + \left(\mathsf{neg}\left(\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) + \left(1 + \left(\mathsf{neg}\left(\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) + \left(1 + \left(\mathsf{neg}\left(\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\right)} \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) + \left(1 - \left(\sqrt{x} + \left(\sqrt{t} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) + \left(1 - \left(\sqrt{x} + \color{blue}{\sqrt{t}}\right)\right) \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 79.0

         \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) + \left(1 - \left(\sqrt{x} + \color{blue}{\sqrt{t}}\right)\right) \]

   8. Simplified 79.0%

      \[\leadsto \left(\sqrt{1 + t} + \left(\sqrt{1 + z} + \sqrt{1 + x}\right)\right) + \left(1 - \left(\sqrt{x} + \color{blue}{\sqrt{t}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 3:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right) + \left(1 - \left(\sqrt{x} + \sqrt{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y} - \sqrt{y}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + z} - \sqrt{z}\\ t_5 := t\_3 + \left(\left(\left(t\_1 - \sqrt{x}\right) + t\_2\right) + t\_4\right)\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_5 \leq 1.99999999995:\\ \;\;\;\;t\_1 + \left(t\_2 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_4 + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_5 (+ t_3 (+ (+ (- t_1 (sqrt x)) t_2) t_4))))
   (if (<= t_5 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_5 1.99999999995)
       (+ t_1 (- t_2 (sqrt x)))
       (+ t_3 (+ t_4 (- (fma x 0.5 2.0) (sqrt y))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + y)) - sqrt(y);
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + z)) - sqrt(z);
	double t_5 = t_3 + (((t_1 - sqrt(x)) + t_2) + t_4);
	double tmp;
	if (t_5 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_5 <= 1.99999999995) {
		tmp = t_1 + (t_2 - sqrt(x));
	} else {
		tmp = t_3 + (t_4 + (fma(x, 0.5, 2.0) - sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_5 = Float64(t_3 + Float64(Float64(Float64(t_1 - sqrt(x)) + t_2) + t_4))
	tmp = 0.0
	if (t_5 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_5 <= 1.99999999995)
		tmp = Float64(t_1 + Float64(t_2 - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(t_4 + Float64(fma(x, 0.5, 2.0) - 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[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[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.99999999995], N[(t$95$1 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$4 + N[(N[(x * 0.5 + 2.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 3.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 35.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 63.6

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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))) < 1.99999999995

   1. Initial program 93.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 19.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 19.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 29.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 29.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   if 1.99999999995 < (+.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 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 66.6

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 66.6%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 40.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 40.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 39.6

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 39.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.99999999995:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\mathsf{fma}\left(x, 0.5, 2\right) - \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := t\_2 - \sqrt{y}\\ t_4 := \sqrt{1 + z}\\ t_5 := \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + \left(t\_4 - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_3 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ (+ (- t_1 (sqrt x)) t_3) (- t_4 (sqrt z))))))
   (if (<= t_5 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_5 2.0)
       (+ t_1 (- t_3 (sqrt x)))
       (- (+ t_4 (+ 1.0 t_2)) (+ (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((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 - sqrt(y);
	double t_4 = sqrt((1.0 + z));
	double t_5 = (sqrt((1.0 + t)) - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_4 - sqrt(z)));
	double tmp;
	if (t_5 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_5 <= 2.0) {
		tmp = t_1 + (t_3 - sqrt(x));
	} else {
		tmp = (t_4 + (1.0 + t_2)) - (sqrt(x) + (sqrt(y) + 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) :: t_5
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_2 - sqrt(y)
    t_4 = sqrt((1.0d0 + z))
    t_5 = (sqrt((1.0d0 + t)) - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_4 - sqrt(z)))
    if (t_5 <= 5d-8) then
        tmp = 0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / x)))
    else if (t_5 <= 2.0d0) then
        tmp = t_1 + (t_3 - sqrt(x))
    else
        tmp = (t_4 + (1.0d0 + t_2)) - (sqrt(x) + (sqrt(y) + 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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = t_2 - Math.sqrt(y);
	double t_4 = Math.sqrt((1.0 + z));
	double t_5 = (Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (((t_1 - Math.sqrt(x)) + t_3) + (t_4 - Math.sqrt(z)));
	double tmp;
	if (t_5 <= 5e-8) {
		tmp = 0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / x)));
	} else if (t_5 <= 2.0) {
		tmp = t_1 + (t_3 - Math.sqrt(x));
	} else {
		tmp = (t_4 + (1.0 + t_2)) - (Math.sqrt(x) + (Math.sqrt(y) + 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((x + 1.0))
	t_2 = math.sqrt((1.0 + y))
	t_3 = t_2 - math.sqrt(y)
	t_4 = math.sqrt((1.0 + z))
	t_5 = (math.sqrt((1.0 + t)) - math.sqrt(t)) + (((t_1 - math.sqrt(x)) + t_3) + (t_4 - math.sqrt(z)))
	tmp = 0
	if t_5 <= 5e-8:
		tmp = 0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / x)))
	elif t_5 <= 2.0:
		tmp = t_1 + (t_3 - math.sqrt(x))
	else:
		tmp = (t_4 + (1.0 + t_2)) - (math.sqrt(x) + (math.sqrt(y) + 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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + Float64(t_4 - sqrt(z))))
	tmp = 0.0
	if (t_5 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_5 <= 2.0)
		tmp = Float64(t_1 + Float64(t_3 - sqrt(x)));
	else
		tmp = Float64(Float64(t_4 + Float64(1.0 + t_2)) - Float64(sqrt(x) + Float64(sqrt(y) + 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((x + 1.0));
	t_2 = sqrt((1.0 + y));
	t_3 = t_2 - sqrt(y);
	t_4 = sqrt((1.0 + z));
	t_5 = (sqrt((1.0 + t)) - sqrt(t)) + (((t_1 - sqrt(x)) + t_3) + (t_4 - sqrt(z)));
	tmp = 0.0;
	if (t_5 <= 5e-8)
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	elseif (t_5 <= 2.0)
		tmp = t_1 + (t_3 - sqrt(x));
	else
		tmp = (t_4 + (1.0 + t_2)) - (sqrt(x) + (sqrt(y) + 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(t$95$1 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 4.9999999999999998e-8

   1. Initial program 5.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 3.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 35.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 63.6

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 63.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 23.0

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 23.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 25.7

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 25.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   if 2 < (+.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 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 24.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 24.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{z}} + \sqrt{y}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 20.6

          \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\\ t_5 := t\_4 + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_5 \leq 2.0002:\\ \;\;\;\;t\_2 + \left(t\_4 + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(1 + \left(t\_3 + \left(t\_1 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_3 (sqrt y))))
        (t_5 (+ t_4 (- t_1 (sqrt z)))))
   (if (<= t_5 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_5 2.0002)
       (+ t_2 (+ t_4 (* 0.5 (sqrt (/ 1.0 z)))))
       (+ t_2 (+ 1.0 (+ t_3 (- t_1 (+ (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 + z));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + y));
	double t_4 = (sqrt((x + 1.0)) - sqrt(x)) + (t_3 - sqrt(y));
	double t_5 = t_4 + (t_1 - sqrt(z));
	double tmp;
	if (t_5 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_5 <= 2.0002) {
		tmp = t_2 + (t_4 + (0.5 * sqrt((1.0 / z))));
	} else {
		tmp = t_2 + (1.0 + (t_3 + (t_1 - (sqrt(x) + (sqrt(y) + 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) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    t_3 = sqrt((1.0d0 + y))
    t_4 = (sqrt((x + 1.0d0)) - sqrt(x)) + (t_3 - sqrt(y))
    t_5 = t_4 + (t_1 - sqrt(z))
    if (t_5 <= 5d-8) then
        tmp = 0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / x)))
    else if (t_5 <= 2.0002d0) then
        tmp = t_2 + (t_4 + (0.5d0 * sqrt((1.0d0 / z))))
    else
        tmp = t_2 + (1.0d0 + (t_3 + (t_1 - (sqrt(x) + (sqrt(y) + 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 + z));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_3 = Math.sqrt((1.0 + y));
	double t_4 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (t_3 - Math.sqrt(y));
	double t_5 = t_4 + (t_1 - Math.sqrt(z));
	double tmp;
	if (t_5 <= 5e-8) {
		tmp = 0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / x)));
	} else if (t_5 <= 2.0002) {
		tmp = t_2 + (t_4 + (0.5 * Math.sqrt((1.0 / z))));
	} else {
		tmp = t_2 + (1.0 + (t_3 + (t_1 - (Math.sqrt(x) + (Math.sqrt(y) + 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 + z))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_3 = math.sqrt((1.0 + y))
	t_4 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (t_3 - math.sqrt(y))
	t_5 = t_4 + (t_1 - math.sqrt(z))
	tmp = 0
	if t_5 <= 5e-8:
		tmp = 0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / x)))
	elif t_5 <= 2.0002:
		tmp = t_2 + (t_4 + (0.5 * math.sqrt((1.0 / z))))
	else:
		tmp = t_2 + (1.0 + (t_3 + (t_1 - (math.sqrt(x) + (math.sqrt(y) + 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 + z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_3 - sqrt(y)))
	t_5 = Float64(t_4 + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_5 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_5 <= 2.0002)
		tmp = Float64(t_2 + Float64(t_4 + Float64(0.5 * sqrt(Float64(1.0 / z)))));
	else
		tmp = Float64(t_2 + Float64(1.0 + Float64(t_3 + Float64(t_1 - Float64(sqrt(x) + Float64(sqrt(y) + 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 + z));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	t_3 = sqrt((1.0 + y));
	t_4 = (sqrt((x + 1.0)) - sqrt(x)) + (t_3 - sqrt(y));
	t_5 = t_4 + (t_1 - sqrt(z));
	tmp = 0.0;
	if (t_5 <= 5e-8)
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	elseif (t_5 <= 2.0002)
		tmp = t_2 + (t_4 + (0.5 * sqrt((1.0 / z))));
	else
		tmp = t_2 + (1.0 + (t_3 + (t_1 - (sqrt(x) + (sqrt(y) + 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 + 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[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0002], N[(t$95$2 + N[(t$95$4 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 + N[(t$95$3 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 4.9999999999999998e-8

   1. Initial program 30.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 4.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 25.6

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 44.5

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 44.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 54.0

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 54.0%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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 98.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate--l+ N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\sqrt{1 + y}} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{\color{blue}{1 + y}} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 96.7

          \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.0002:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{if}\;t\_2 \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_2 \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)))
        (t_2
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+
           (+ t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
           (- (sqrt (+ 1.0 z)) (sqrt z))))))
   (if (<= t_2 0.2)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_2 1.5) (fma 0.5 (+ x (sqrt (/ 1.0 y))) (- 1.0 (sqrt x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double t_2 = (sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + (sqrt((1.0 + y)) - sqrt(y))) + (sqrt((1.0 + z)) - sqrt(z)));
	double tmp;
	if (t_2 <= 0.2) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_2 <= 1.5) {
		tmp = fma(0.5, (x + sqrt((1.0 / y))), (1.0 - sqrt(x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_2 <= 1.5)
		tmp = fma(0.5, Float64(x + sqrt(Float64(1.0 / y))), Float64(1.0 - sqrt(x)));
	else
		tmp = t_1;
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.2], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.5], N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 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))) < 0.20000000000000001

   1. Initial program 19.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 9.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 36.4

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 36.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 57.0

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 57.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 0.20000000000000001 < (+.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))) < 1.5

   1. Initial program 93.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 29.3

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 29.3%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\left(\left(\color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 29.7

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 29.7%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 29.7

          \[\leadsto \left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 29.7%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x} \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + \left(1 - \sqrt{x}\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, \color{blue}{1 - \sqrt{x}}\right) \]

       8. sqrt-lowering-sqrt.f64 29.4

          \[\leadsto \mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \color{blue}{\sqrt{x}}\right) \]

   14. Simplified 29.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

   if 1.5 < (+.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 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. 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 97.6

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 27.5

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   7. Simplified 27.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 13.1

         \[\leadsto \sqrt{1 + x} + \left(-\color{blue}{\sqrt{x}}\right) \]

   10. Simplified 13.1%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y} - \sqrt{y}\\ t_4 := \left(\left(t\_2 - \sqrt{x}\right) + t\_3\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_2 + \left(t\_3 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + \mathsf{fma}\left(x, 0.5, 2\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_3 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_4 (+ (+ (- t_2 (sqrt x)) t_3) (- t_1 (sqrt z)))))
   (if (<= t_4 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_4 2.0)
       (+ t_2 (- t_3 (sqrt x)))
       (- (- (+ t_1 (fma x 0.5 2.0)) (+ (sqrt x) (sqrt z))) (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((x + 1.0));
	double t_3 = sqrt((1.0 + y)) - sqrt(y);
	double t_4 = ((t_2 - sqrt(x)) + t_3) + (t_1 - sqrt(z));
	double tmp;
	if (t_4 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_4 <= 2.0) {
		tmp = t_2 + (t_3 - sqrt(x));
	} else {
		tmp = ((t_1 + fma(x, 0.5, 2.0)) - (sqrt(x) + sqrt(z))) - 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 = sqrt(Float64(x + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(x)) + t_3) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_4 <= 2.0)
		tmp = Float64(t_2 + Float64(t_3 - sqrt(x)));
	else
		tmp = Float64(Float64(Float64(t_1 + fma(x, 0.5, 2.0)) - Float64(sqrt(x) + sqrt(z))) - 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(t$95$2 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + N[(x * 0.5 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $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))) < 4.9999999999999998e-8

   1. Initial program 30.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 4.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 25.6

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 44.5

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 44.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 29.1

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 29.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 23.4

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 23.4%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   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 95.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.2

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 95.2%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 88.9

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 88.9%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 0.5, 2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \sqrt{x}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

       3. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}} \]

       4. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} - \sqrt{y} \]

       5. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) - \sqrt{y}} \]

   11. Simplified 44.4%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \sqrt{1 + z}\right) - \left(\sqrt{z} + \sqrt{x}\right)\right) - \sqrt{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} + \mathsf{fma}\left(x, 0.5, 2\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) - \sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y}\\ t_4 := \left(t\_2 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\\ \mathbf{if}\;t\_4 \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_4 \leq 1.9999:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + t\_3}\right) - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\left(1 + t\_2\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 t)) (sqrt t)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (+ (- t_2 (sqrt x)) (- t_3 (sqrt y)))))
   (if (<= t_4 0.2)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_4 1.9999)
       (+ (- (+ (fma x 0.5 1.0) (/ 1.0 (+ (sqrt y) t_3))) (sqrt x)) t_1)
       (+
        t_1
        (-
         (+ (+ 1.0 t_2) (fma y 0.5 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 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 + t)) - sqrt(t);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + y));
	double t_4 = (t_2 - sqrt(x)) + (t_3 - sqrt(y));
	double tmp;
	if (t_4 <= 0.2) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_4 <= 1.9999) {
		tmp = ((fma(x, 0.5, 1.0) + (1.0 / (sqrt(y) + t_3))) - sqrt(x)) + t_1;
	} else {
		tmp = t_1 + (((1.0 + t_2) + fma(y, 0.5, (1.0 / (sqrt(z) + sqrt((1.0 + 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 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(Float64(t_2 - sqrt(x)) + Float64(t_3 - sqrt(y)))
	tmp = 0.0
	if (t_4 <= 0.2)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_4 <= 1.9999)
		tmp = Float64(Float64(Float64(fma(x, 0.5, 1.0) + Float64(1.0 / Float64(sqrt(y) + t_3))) - sqrt(x)) + t_1);
	else
		tmp = Float64(t_1 + Float64(Float64(Float64(1.0 + t_2) + fma(y, 0.5, Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + 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[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.2], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.9999], N[(N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(y * 0.5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.20000000000000001

   1. Initial program 76.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 5.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 13.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 13.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 18.1

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 18.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 0.20000000000000001 < (+.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))) < 1.99990000000000001

   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 53.3

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 53.3%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\sqrt{\color{blue}{1 + y}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\sqrt{1 + y} \cdot \sqrt{\color{blue}{1 + y}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-lowering-+.f64 53.7

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 53.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 31.0

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 31.0%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.99990000000000001 < (+.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 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. 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 97.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{\left(\left(\left(1 + \sqrt{1 + x}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.9999:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 + \sqrt{x + 1}\right) + \mathsf{fma}\left(y, 0.5, \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := t\_2 - \sqrt{y}\\ t_4 := \left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;t\_1 + \left(t\_3 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_2\right) + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (+ (+ (- t_1 (sqrt x)) t_3) (- (sqrt (+ 1.0 z)) (sqrt z)))))
   (if (<= t_4 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_4 2.5)
       (+ t_1 (- t_3 (sqrt x)))
       (+ (+ 1.0 t_2) (- 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((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 - sqrt(y);
	double t_4 = ((t_1 - sqrt(x)) + t_3) + (sqrt((1.0 + z)) - sqrt(z));
	double tmp;
	if (t_4 <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_4 <= 2.5) {
		tmp = t_1 + (t_3 - sqrt(x));
	} else {
		tmp = (1.0 + t_2) + (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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_2 - sqrt(y)
    t_4 = ((t_1 - sqrt(x)) + t_3) + (sqrt((1.0d0 + z)) - sqrt(z))
    if (t_4 <= 5d-8) then
        tmp = 0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / x)))
    else if (t_4 <= 2.5d0) then
        tmp = t_1 + (t_3 - sqrt(x))
    else
        tmp = (1.0d0 + t_2) + (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((x + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = t_2 - Math.sqrt(y);
	double t_4 = ((t_1 - Math.sqrt(x)) + t_3) + (Math.sqrt((1.0 + z)) - Math.sqrt(z));
	double tmp;
	if (t_4 <= 5e-8) {
		tmp = 0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / x)));
	} else if (t_4 <= 2.5) {
		tmp = t_1 + (t_3 - Math.sqrt(x));
	} else {
		tmp = (1.0 + t_2) + (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((x + 1.0))
	t_2 = math.sqrt((1.0 + y))
	t_3 = t_2 - math.sqrt(y)
	t_4 = ((t_1 - math.sqrt(x)) + t_3) + (math.sqrt((1.0 + z)) - math.sqrt(z))
	tmp = 0
	if t_4 <= 5e-8:
		tmp = 0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / x)))
	elif t_4 <= 2.5:
		tmp = t_1 + (t_3 - math.sqrt(x))
	else:
		tmp = (1.0 + t_2) + (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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_4 <= 2.5)
		tmp = Float64(t_1 + Float64(t_3 - sqrt(x)));
	else
		tmp = Float64(Float64(1.0 + t_2) + Float64(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((x + 1.0));
	t_2 = sqrt((1.0 + y));
	t_3 = t_2 - sqrt(y);
	t_4 = ((t_1 - sqrt(x)) + t_3) + (sqrt((1.0 + z)) - sqrt(z));
	tmp = 0.0;
	if (t_4 <= 5e-8)
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	elseif (t_4 <= 2.5)
		tmp = t_1 + (t_3 - sqrt(x));
	else
		tmp = (1.0 + t_2) + (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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(t$95$1 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[x], $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))) < 4.9999999999999998e-8

   1. Initial program 30.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 4.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 25.6

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 44.5

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 44.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (+.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 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 29.5

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 29.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 23.8

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 23.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

   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 98.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 43.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 43.7

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 43.7%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{1}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 43.6%

       \[\leadsto \left(\sqrt{1 + y} + \color{blue}{1}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.5:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{x + 1} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + y}\\ t_3 := \left(t\_1 - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\\ \mathbf{if}\;t\_3 \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;t\_3 \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_2 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (+ (- t_1 (sqrt x)) (- t_2 (sqrt y)))))
   (if (<= t_3 0.2)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (if (<= t_3 1.002)
       (fma 0.5 (+ x (sqrt (/ 1.0 y))) (- 1.0 (sqrt x)))
       (+ t_1 (- t_2 (+ (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((x + 1.0));
	double t_2 = sqrt((1.0 + y));
	double t_3 = (t_1 - sqrt(x)) + (t_2 - sqrt(y));
	double tmp;
	if (t_3 <= 0.2) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else if (t_3 <= 1.002) {
		tmp = fma(0.5, (x + sqrt((1.0 / y))), (1.0 - sqrt(x)));
	} else {
		tmp = t_1 + (t_2 - (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(x + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(Float64(t_1 - sqrt(x)) + Float64(t_2 - sqrt(y)))
	tmp = 0.0
	if (t_3 <= 0.2)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	elseif (t_3 <= 1.002)
		tmp = fma(0.5, Float64(x + sqrt(Float64(1.0 / y))), Float64(1.0 - sqrt(x)));
	else
		tmp = Float64(t_1 + Float64(t_2 - 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.2], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.002], N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$2 - 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))) < 0.20000000000000001

   1. Initial program 76.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 5.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 13.3

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 13.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 18.1

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 18.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 0.20000000000000001 < (+.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))) < 1.002

   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

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 51.5

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.5%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\left(\left(\color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 52.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 52.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 30.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 30.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x} \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + \left(1 - \sqrt{x}\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, \color{blue}{1 - \sqrt{x}}\right) \]

       8. sqrt-lowering-sqrt.f64 20.9

          \[\leadsto \mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \color{blue}{\sqrt{x}}\right) \]

   14. Simplified 20.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

   if 1.002 < (+.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 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 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 26.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 26.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 36.3

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right) \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x))))
   (if (<=
        (+
         (- (sqrt (+ 1.0 t)) (sqrt t))
         (+
          (+ t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
          (- (sqrt (+ 1.0 z)) (sqrt z))))
        1.5)
     (fma 0.5 (+ x (sqrt (/ 1.0 y))) (- 1.0 (sqrt x)))
     t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((t_1 + (sqrt((1.0 + y)) - sqrt(y))) + (sqrt((1.0 + z)) - sqrt(z)))) <= 1.5) {
		tmp = fma(0.5, (x + sqrt((1.0 / y))), (1.0 - sqrt(x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)))) <= 1.5)
		tmp = fma(0.5, Float64(x + sqrt(Float64(1.0 / y))), Float64(1.0 - sqrt(x)));
	else
		tmp = t_1;
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 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))) < 1.5

   1. Initial program 80.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 25.0

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 25.0%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \left(\left(\left(\color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 25.3

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 25.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. sqrt-lowering-sqrt.f64 25.3

          \[\leadsto \left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Simplified 25.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x} \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right) + \left(1 - \sqrt{x}\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1 - \sqrt{x}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, \color{blue}{1 - \sqrt{x}}\right) \]

       8. sqrt-lowering-sqrt.f64 25.1

          \[\leadsto \mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \color{blue}{\sqrt{x}}\right) \]

   14. Simplified 25.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)} \]

   if 1.5 < (+.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 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. 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 97.6

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 27.5

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   7. Simplified 27.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 13.1

         \[\leadsto \sqrt{1 + x} + \left(-\color{blue}{\sqrt{x}}\right) \]

   10. Simplified 13.1%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 70.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t\_1 - \sqrt{x} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ x 1.0))))
   (if (<= (- t_1 (sqrt x)) 5e-8)
     (* 0.5 (+ (sqrt (/ 1.0 t)) (sqrt (/ 1.0 x))))
     (+ t_1 (- (- (sqrt (+ 1.0 y)) (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((x + 1.0));
	double tmp;
	if ((t_1 - sqrt(x)) <= 5e-8) {
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	} else {
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - 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) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if ((t_1 - sqrt(x)) <= 5d-8) then
        tmp = 0.5d0 * (sqrt((1.0d0 / t)) + sqrt((1.0d0 / x)))
    else
        tmp = t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - 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((x + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 5e-8) {
		tmp = 0.5 * (Math.sqrt((1.0 / t)) + Math.sqrt((1.0 / x)));
	} else {
		tmp = t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - 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((x + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 5e-8:
		tmp = 0.5 * (math.sqrt((1.0 / t)) + math.sqrt((1.0 / x)))
	else:
		tmp = t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 5e-8)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / t)) + sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - 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((x + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 5e-8)
		tmp = 0.5 * (sqrt((1.0 / t)) + sqrt((1.0 / x)));
	else
		tmp = t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 5e-8], N[(0.5 * N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 4.9999999999999998e-8

   1. Initial program 88.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \sqrt{1 + x}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)}\right) + \sqrt{1 + x} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \sqrt{x}\right)} + \sqrt{1 + x} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   5. Simplified 19.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{t}}, \sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

      3. /-lowering-/.f64 9.4

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   8. Simplified 9.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} - \left(\sqrt{x} - \sqrt{1 + x}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{t}}} + \sqrt{\frac{1}{x}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{t}} + \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

       7. /-lowering-/.f64 12.2

          \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

   11. Simplified 12.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)} \]

   if 4.9999999999999998e-8 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 38.3

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 38.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      6. associate--r+ N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 36.5

          \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 36.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{t}} + \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt((x + 1.0)) - 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 = sqrt((x + 1.0d0)) - 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 Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((x + 1.0)) - 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[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   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 93.4

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied egg-rr 93.4%

   \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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. +-lowering-+.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

   15. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

   16. sqrt-lowering-sqrt.f64 27.0

       \[\leadsto \sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

7. Simplified 27.0%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{1 + x} + \color{blue}{-1 \cdot \sqrt{x}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

   3. sqrt-lowering-sqrt.f64 16.0

      \[\leadsto \sqrt{1 + x} + \left(-\color{blue}{\sqrt{x}}\right) \]

10. Simplified 16.0%

    \[\leadsto \sqrt{1 + x} + \color{blue}{\left(-\sqrt{x}\right)} \]

11. Final simplification 16.0%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]
12. Add Preprocessing

Alternative 22: 7.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{y}} \end{array} \]

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 y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / y));
}

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 / y))
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 / y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / y)))
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 / y));
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 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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 55.3

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 55.3%

   \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in y around inf

   \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \left(\left(\left(\color{blue}{\frac{1}{2} \cdot \left(x + \sqrt{\frac{1}{y}}\right)} + 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\frac{1}{y}}, 1\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{x + \sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \color{blue}{\sqrt{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x + \sqrt{\color{blue}{\frac{1}{y}}}, 1\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. sqrt-lowering-sqrt.f64 29.0

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Simplified 29.0%

   \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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 7.3

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{y}}} \]

11. Simplified 7.3%

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{y}}} \]
12. Add Preprocessing

Alternative 23: 7.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt z))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(z);
}

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 = sqrt(z)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(z);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(z)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(z)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(z);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[z], $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in 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 18.3

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 18.3%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 18.2

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

8. Simplified 18.2%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\sqrt{z}} \]
10. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 6.8

       \[\leadsto \color{blue}{\sqrt{z}} \]

11. Simplified 6.8%

    \[\leadsto \color{blue}{\sqrt{z}} \]
12. Add Preprocessing

Developer Target 1: 99.3% 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 2024204 
(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))))