Main:z from

Percentage Accurate: 91.9% → 98.3%

Time: 27.1s

Alternatives: 23

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% 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: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + t\_3}\right)\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + t\_2} + \left(\frac{1}{\sqrt{y} + t\_1} + t\_3\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(t\_1 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 z)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 0.9999999989571969)
     (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) t_3)))
     (if (<= t_4 2.5)
       (-
        (+ (/ 1.0 (+ (sqrt z) t_2)) (+ (/ 1.0 (+ (sqrt y) t_1)) t_3))
        (sqrt x))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (+ 1.0 (+ t_1 (- 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((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((x + 1.0));
	double t_4 = (t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 0.9999999989571969) {
		tmp = fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + t_3)));
	} else if (t_4 <= 2.5) {
		tmp = ((1.0 / (sqrt(z) + t_2)) + ((1.0 / (sqrt(y) + t_1)) + t_3)) - sqrt(x);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (1.0 + (t_1 + (t_2 - (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 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 0.9999999989571969)
		tmp = fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + t_3)));
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + t_3)) - sqrt(x));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 + Float64(t_1 + Float64(t_2 - 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[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.9999999989571969], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 + N[(t$95$2 - 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))) < 0.999999998957196934

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 25.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 48.0

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

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 54.0%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 27.3

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

   13. Simplified 27.3%

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

   if 0.999999998957196934 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 97.1%

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

      1. flip-- N/A

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

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

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

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

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

      12. sqrt-lowering-sqrt.f64 97.4

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

   4. Applied egg-rr 97.4%

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

      1. flip-- N/A

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

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

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 98.0

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in t around inf

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

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

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

   9. Simplified 31.7%

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

   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 99.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{x + 1}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{1 + z}\\ t_6 := \left(t\_3 - \sqrt{t}\right) + \left(\left(t\_5 - \sqrt{z}\right) + \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_1, \frac{1}{\sqrt{x} + t\_2}\right)\\ \mathbf{elif}\;t\_6 \leq 2.0001:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, t\_1, t\_4\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 3.5:\\ \;\;\;\;\left(\left(t\_5 + 2\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(t\_4 + t\_3\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (sqrt (+ 1.0 z)))
        (t_6
         (+
          (- t_3 (sqrt t))
          (+ (- t_5 (sqrt z)) (+ (- t_2 (sqrt x)) (- t_4 (sqrt y)))))))
   (if (<= t_6 0.9999999989571969)
     (fma 0.5 t_1 (/ 1.0 (+ (sqrt x) t_2)))
     (if (<= t_6 2.0001)
       (+ t_2 (- (fma 0.5 t_1 t_4) (+ (sqrt x) (sqrt y))))
       (if (<= t_6 3.5)
         (- (+ (+ t_5 2.0) (/ z (- (sqrt y) (sqrt z)))) (sqrt x))
         (+
          2.0
          (- (+ t_4 t_3) (+ (sqrt t) (+ (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((x + 1.0));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt((1.0 + z));
	double t_6 = (t_3 - sqrt(t)) + ((t_5 - sqrt(z)) + ((t_2 - sqrt(x)) + (t_4 - sqrt(y))));
	double tmp;
	if (t_6 <= 0.9999999989571969) {
		tmp = fma(0.5, t_1, (1.0 / (sqrt(x) + t_2)));
	} else if (t_6 <= 2.0001) {
		tmp = t_2 + (fma(0.5, t_1, t_4) - (sqrt(x) + sqrt(y)));
	} else if (t_6 <= 3.5) {
		tmp = ((t_5 + 2.0) + (z / (sqrt(y) - sqrt(z)))) - sqrt(x);
	} else {
		tmp = 2.0 + ((t_4 + t_3) - (sqrt(t) + (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 / z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = sqrt(Float64(1.0 + z))
	t_6 = Float64(Float64(t_3 - sqrt(t)) + Float64(Float64(t_5 - sqrt(z)) + Float64(Float64(t_2 - sqrt(x)) + Float64(t_4 - sqrt(y)))))
	tmp = 0.0
	if (t_6 <= 0.9999999989571969)
		tmp = fma(0.5, t_1, Float64(1.0 / Float64(sqrt(x) + t_2)));
	elseif (t_6 <= 2.0001)
		tmp = Float64(t_2 + Float64(fma(0.5, t_1, t_4) - Float64(sqrt(x) + sqrt(y))));
	elseif (t_6 <= 3.5)
		tmp = Float64(Float64(Float64(t_5 + 2.0) + Float64(z / Float64(sqrt(y) - sqrt(z)))) - sqrt(x));
	else
		tmp = Float64(2.0 + Float64(Float64(t_4 + t_3) - Float64(sqrt(t) + 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 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.9999999989571969], N[(0.5 * t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0001], N[(t$95$2 + N[(N[(0.5 * t$95$1 + t$95$4), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.5], N[(N[(N[(t$95$5 + 2.0), $MachinePrecision] + N[(z / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $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))) < 0.999999998957196934

   1. Initial program 28.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 21.1

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

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 31.7%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 48.8

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

   13. Simplified 48.8%

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

   if 0.999999998957196934 < (+.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))) < 2.00010000000000021

   1. Initial program 96.3%

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

   3. 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 17.2

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

   5. Simplified 17.2%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 21.7

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

   8. Simplified 21.7%

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

   if 2.00010000000000021 < (+.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))) < 3.5

   1. Initial program 98.2%

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

   3. Taylor expanded in 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 25.8

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

   5. Simplified 25.8%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. rem-square-sqrt N/A

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

      5. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 26.0

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

   7. Applied egg-rr 26.0%

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

   8. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 43.4

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

   10. Simplified 43.4%

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

   11. Taylor expanded in x around 0

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

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

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

       2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       11. sqrt-lowering-sqrt.f64 32.7

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

   13. Simplified 32.7%

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

   if 3.5 < (+.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 99.8%

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

   3. Taylor expanded in z around 0

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

      1. associate--r+ N/A

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

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

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

      3. associate-+r+ N/A

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

      4. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sqrt-lowering-sqrt.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

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

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

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

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      17. sqrt-lowering-sqrt.f64 98.9

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

   8. Simplified 98.9%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0001:\\ \;\;\;\;\sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3.5:\\ \;\;\;\;\left(\left(\sqrt{1 + z} + 2\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + y} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + z}\\ t_5 := \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_2 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_1, \frac{1}{\sqrt{x} + t\_2}\right)\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, t\_1, t\_3\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_4 + 2\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ (- t_4 (sqrt z)) (+ (- t_2 (sqrt x)) (- t_3 (sqrt y)))))))
   (if (<= t_5 0.9999999989571969)
     (fma 0.5 t_1 (/ 1.0 (+ (sqrt x) t_2)))
     (if (<= t_5 2.0001)
       (+ t_2 (- (fma 0.5 t_1 t_3) (+ (sqrt x) (sqrt y))))
       (- (+ (+ t_4 2.0) (/ z (- (sqrt y) (sqrt z)))) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((1.0 + z));
	double t_5 = (sqrt((1.0 + t)) - sqrt(t)) + ((t_4 - sqrt(z)) + ((t_2 - sqrt(x)) + (t_3 - sqrt(y))));
	double tmp;
	if (t_5 <= 0.9999999989571969) {
		tmp = fma(0.5, t_1, (1.0 / (sqrt(x) + t_2)));
	} else if (t_5 <= 2.0001) {
		tmp = t_2 + (fma(0.5, t_1, t_3) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = ((t_4 + 2.0) + (z / (sqrt(y) - sqrt(z)))) - sqrt(x);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_2 - sqrt(x)) + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_5 <= 0.9999999989571969)
		tmp = fma(0.5, t_1, Float64(1.0 / Float64(sqrt(x) + t_2)));
	elseif (t_5 <= 2.0001)
		tmp = Float64(t_2 + Float64(fma(0.5, t_1, t_3) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(Float64(t_4 + 2.0) + Float64(z / Float64(sqrt(y) - sqrt(z)))) - sqrt(x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $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[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[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.9999999989571969], N[(0.5 * t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(t$95$2 + N[(N[(0.5 * t$95$1 + t$95$3), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 + 2.0), $MachinePrecision] + N[(z / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $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))) < 0.999999998957196934

   1. Initial program 28.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 21.1

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

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 31.7%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 48.8

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

   13. Simplified 48.8%

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

   if 0.999999998957196934 < (+.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))) < 2.00010000000000021

   1. Initial program 96.3%

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

   3. 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 17.2

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

   5. Simplified 17.2%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 21.7

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

   8. Simplified 21.7%

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

   if 2.00010000000000021 < (+.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 98.6%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. rem-square-sqrt N/A

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

      5. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 24.7

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

   7. Applied egg-rr 24.7%

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

   8. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 38.3

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

   10. Simplified 38.3%

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

   11. Taylor expanded in x around 0

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

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

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

       2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       11. sqrt-lowering-sqrt.f64 30.0

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

   13. Simplified 30.0%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0001:\\ \;\;\;\;\sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} + 2\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 - \sqrt{x}\\ t_3 := \sqrt{1 + y}\\ t_4 := \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_2 + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_4 \leq 1.0005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;\left(t\_3 + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + 2\\ \end{array} \end{array} \]

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 (- t_1 (sqrt x)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ t_2 (- t_3 (sqrt y)))))))
   (if (<= t_4 1.0005)
     (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
     (if (<= t_4 2.5)
       (- (+ t_3 (fma x 0.5 1.0)) (+ (sqrt x) (sqrt y)))
       (+ t_2 2.0)))))

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 = t_1 - sqrt(x);
	double t_3 = sqrt((1.0 + y));
	double t_4 = (sqrt((1.0 + t)) - sqrt(t)) + ((sqrt((1.0 + z)) - sqrt(z)) + (t_2 + (t_3 - sqrt(y))));
	double tmp;
	if (t_4 <= 1.0005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else if (t_4 <= 2.5) {
		tmp = (t_3 + fma(x, 0.5, 1.0)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = t_2 + 2.0;
	}
	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(t_1 - sqrt(x))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_2 + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_4 <= 1.0005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(t_3 + fma(x, 0.5, 1.0)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(t_2 + 2.0);
	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[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(t$95$3 + N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + 2.0), $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))) < 1.00049999999999994

   1. Initial program 79.2%

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

   3. Taylor expanded in 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 4.8

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

   5. Simplified 4.8%

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

   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 21.9

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

   8. Simplified 21.9%

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

   9. Taylor expanded in y around inf

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

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

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

       2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

       8. sqrt-lowering-sqrt.f64 23.2

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

   11. Simplified 23.2%

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

   if 1.00049999999999994 < (+.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))) < 2.5

   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 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 23.7

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

   5. Simplified 23.7%

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

   6. Taylor expanded in 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 20.6

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

   8. Simplified 20.6%

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

   9. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. associate-+r+ N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

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

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

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

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

       13. sqrt-lowering-sqrt.f64 16.2

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

   11. Simplified 16.2%

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

   if 2.5 < (+.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 99.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 25.0

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

   5. Simplified 25.0%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. rem-square-sqrt N/A

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

      5. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 25.2

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 37.9

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

   10. Simplified 37.9%

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

   11. Taylor expanded in z around 0

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

       1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

       6. sqrt-lowering-sqrt.f64 61.7

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

   13. Simplified 61.7%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.0005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.5:\\ \;\;\;\;\left(\sqrt{1 + y} + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_3, \frac{1}{\sqrt{x} + t\_4}\right)\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;t\_4 + \left(\mathsf{fma}\left(0.5, t\_3, t\_1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(t\_1 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 z)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_5 0.9999999989571969)
     (fma 0.5 t_3 (/ 1.0 (+ (sqrt x) t_4)))
     (if (<= t_5 2.0001)
       (+ t_4 (- (fma 0.5 t_3 t_1) (+ (sqrt x) (sqrt y))))
       (+
        (- (sqrt (+ 1.0 t)) (sqrt t))
        (+ 1.0 (+ t_1 (- 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((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((x + 1.0));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_5 <= 0.9999999989571969) {
		tmp = fma(0.5, t_3, (1.0 / (sqrt(x) + t_4)));
	} else if (t_5 <= 2.0001) {
		tmp = t_4 + (fma(0.5, t_3, t_1) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (1.0 + (t_1 + (t_2 - (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 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_5 <= 0.9999999989571969)
		tmp = fma(0.5, t_3, Float64(1.0 / Float64(sqrt(x) + t_4)));
	elseif (t_5 <= 2.0001)
		tmp = Float64(t_4 + Float64(fma(0.5, t_3, t_1) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 + Float64(t_1 + Float64(t_2 - 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[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.9999999989571969], N[(0.5 * t$95$3 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(t$95$4 + N[(N[(0.5 * t$95$3 + t$95$1), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 + N[(t$95$2 - 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))) < 0.999999998957196934

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 25.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 48.0

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

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 54.0%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 27.3

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

   13. Simplified 27.3%

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

   if 0.999999998957196934 < (+.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))) < 2.00010000000000021

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 16.7

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

   5. Simplified 16.7%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 21.4

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

   8. Simplified 21.4%

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

   if 2.00010000000000021 < (+.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 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.8

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

   5. Simplified 95.8%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.0001:\\ \;\;\;\;\sqrt{x + 1} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.5× speedup

Math

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

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

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 35.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 38.8

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

   8. Simplified 38.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 41.2%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 25.3

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

   13. Simplified 25.3%

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

   if 1 < (+.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 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 24.8

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

   5. Simplified 24.8%

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

   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 22.5

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

   8. Simplified 22.5%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 18.1

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

   10. Applied egg-rr 18.1%

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

   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 99.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 51.3

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

   5. Simplified 51.3%

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

   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. +-lowering-+.f64 N/A

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

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

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

      13. sqrt-lowering-sqrt.f64 48.1

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

   8. Simplified 48.1%

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

4. Final simplification 24.6%

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

Alternative 7: 89.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

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

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

   5. Simplified 35.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 38.8

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

   8. Simplified 38.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

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

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

   10. Applied egg-rr 41.2%

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

   11. Taylor expanded in t around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 25.3

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

   13. Simplified 25.3%

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

   if 1 < (+.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 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 24.8

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

   5. Simplified 24.8%

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

   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 22.5

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

   8. Simplified 22.5%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

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

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

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

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x} \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x} \]

      13. sqrt-lowering-sqrt.f64 18.1

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}} \]

   10. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x}} \]

   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 99.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 51.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 51.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 51.3%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 48.7

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 48.7%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}} \]
   12. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}} \]

       2. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right)} - \sqrt{x} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right)} - \sqrt{x} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + z}\right)} + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + z}}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + z}}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + z}\right) + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right) - \sqrt{x} \]

       8. --lowering--.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right) - \sqrt{x} \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right) - \sqrt{x} \]

       10. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right) - \sqrt{x} \]

       11. sqrt-lowering-sqrt.f64 45.5

           \[\leadsto \left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \color{blue}{\sqrt{x}} \]

   13. Simplified 45.5%

       \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} + 2\right) + \frac{z}{\sqrt{y} - \sqrt{z}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{x + 1}\\ t_4 := t\_2 + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + t\_3}\right)\\ \mathbf{elif}\;t\_4 \leq 1.9999999999999998:\\ \;\;\;\;\left(\left(t\_1 + t\_3\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(t\_2 + t\_3\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ t_2 (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 1.0)
     (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) t_3)))
     (if (<= t_4 1.9999999999999998)
       (- (- (+ t_1 t_3) (sqrt y)) (sqrt x))
       (+ 1.0 (- (+ t_2 t_3) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_2 + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 1.0) {
		tmp = fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + t_3)));
	} else if (t_4 <= 1.9999999999999998) {
		tmp = ((t_1 + t_3) - sqrt(y)) - sqrt(x);
	} else {
		tmp = 1.0 + ((t_2 + t_3) - sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_2 + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + t_3)));
	elseif (t_4 <= 1.9999999999999998)
		tmp = Float64(Float64(Float64(t_1 + t_3) - sqrt(y)) - sqrt(x));
	else
		tmp = Float64(1.0 + Float64(Float64(t_2 + t_3) - sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.9999999999999998], N[(N[(N[(t$95$1 + t$95$3), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(t$95$2 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1

   1. Initial program 86.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 35.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 38.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 38.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. div-inv N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}, \frac{1}{\sqrt{x + 1} + \sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   10. Applied egg-rr 41.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) \]

       8. +-lowering-+.f64 25.3

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) \]

   13. Simplified 25.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

   if 1 < (+.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))) < 1.99999999999999978

   1. Initial program 91.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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 15.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 15.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in 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 15.9

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 15.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x} \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x} \]

      13. sqrt-lowering-sqrt.f64 10.8

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}} \]

   10. Applied egg-rr 10.8%

       \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x}} \]

   if 1.99999999999999978 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 34.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 34.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 34.4%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 34.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{-1 \cdot \sqrt{z}}\right)\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)}\right)\right) - \sqrt{x}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)}\right)\right) - \sqrt{x}\right) \]

       3. sqrt-lowering-sqrt.f64 41.1

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(-\color{blue}{\sqrt{z}}\right)\right)\right) - \sqrt{x}\right) \]

   13. Simplified 41.1%

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(-\sqrt{z}\right)}\right)\right) - \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.9999999999999998:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{x + 1}\\ t_4 := t\_2 + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 1.0005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 1.9999999999999998:\\ \;\;\;\;\left(\left(t\_1 + t\_3\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(t\_2 + t\_3\right) - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ t_2 (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 1.0005)
     (- (fma 0.5 (sqrt (/ 1.0 y)) t_3) (sqrt x))
     (if (<= t_4 1.9999999999999998)
       (- (- (+ t_1 t_3) (sqrt y)) (sqrt x))
       (+ 1.0 (- (+ t_2 t_3) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_2 + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 1.0005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_3) - sqrt(x);
	} else if (t_4 <= 1.9999999999999998) {
		tmp = ((t_1 + t_3) - sqrt(y)) - sqrt(x);
	} else {
		tmp = 1.0 + ((t_2 + t_3) - sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_2 + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 1.0005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_3) - sqrt(x));
	elseif (t_4 <= 1.9999999999999998)
		tmp = Float64(Float64(Float64(t_1 + t_3) - sqrt(y)) - sqrt(x));
	else
		tmp = Float64(1.0 + Float64(Float64(t_2 + t_3) - sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.9999999999999998], N[(N[(N[(t$95$1 + t$95$3), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(t$95$2 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00049999999999994

   1. Initial program 85.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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 4.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 4.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in 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 17.0

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 17.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       8. sqrt-lowering-sqrt.f64 17.4

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 17.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00049999999999994 < (+.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))) < 1.99999999999999978

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 17.8

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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 17.7

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right)} - \sqrt{x} \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x} \]

      13. sqrt-lowering-sqrt.f64 11.8

          \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}} \]

   10. Applied egg-rr 11.8%

       \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x}} \]

   if 1.99999999999999978 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 34.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 34.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 34.4%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 34.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{-1 \cdot \sqrt{z}}\right)\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)}\right)\right) - \sqrt{x}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{z}\right)\right)}\right)\right) - \sqrt{x}\right) \]

       3. sqrt-lowering-sqrt.f64 41.1

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \left(-\color{blue}{\sqrt{z}}\right)\right)\right) - \sqrt{x}\right) \]

   13. Simplified 41.1%

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\left(-\sqrt{z}\right)}\right)\right) - \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.0005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.9999999999999998:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + \sqrt{x + 1}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 - \sqrt{x}\\ t_3 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_2 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_3 \leq 1.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2.5:\\ \;\;\;\;1 + \left(\left(t\_1 + z \cdot 0\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + 2\\ \end{array} \end{array} \]

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 (- t_1 (sqrt x)))
        (t_3
         (+
          (- (sqrt (+ 1.0 z)) (sqrt z))
          (+ t_2 (- (sqrt (+ 1.0 y)) (sqrt y))))))
   (if (<= t_3 1.5)
     t_2
     (if (<= t_3 2.5) (+ 1.0 (- (+ t_1 (* z 0.0)) (sqrt x))) (+ t_2 2.0)))))

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 = t_1 - sqrt(x);
	double t_3 = (sqrt((1.0 + z)) - sqrt(z)) + (t_2 + (sqrt((1.0 + y)) - sqrt(y)));
	double tmp;
	if (t_3 <= 1.5) {
		tmp = t_2;
	} else if (t_3 <= 2.5) {
		tmp = 1.0 + ((t_1 + (z * 0.0)) - sqrt(x));
	} else {
		tmp = t_2 + 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = t_1 - sqrt(x)
    t_3 = (sqrt((1.0d0 + z)) - sqrt(z)) + (t_2 + (sqrt((1.0d0 + y)) - sqrt(y)))
    if (t_3 <= 1.5d0) then
        tmp = t_2
    else if (t_3 <= 2.5d0) then
        tmp = 1.0d0 + ((t_1 + (z * 0.0d0)) - sqrt(x))
    else
        tmp = t_2 + 2.0d0
    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 = t_1 - Math.sqrt(x);
	double t_3 = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (t_2 + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	double tmp;
	if (t_3 <= 1.5) {
		tmp = t_2;
	} else if (t_3 <= 2.5) {
		tmp = 1.0 + ((t_1 + (z * 0.0)) - Math.sqrt(x));
	} else {
		tmp = t_2 + 2.0;
	}
	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 = t_1 - math.sqrt(x)
	t_3 = (math.sqrt((1.0 + z)) - math.sqrt(z)) + (t_2 + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	tmp = 0
	if t_3 <= 1.5:
		tmp = t_2
	elif t_3 <= 2.5:
		tmp = 1.0 + ((t_1 + (z * 0.0)) - math.sqrt(x))
	else:
		tmp = t_2 + 2.0
	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(t_1 - sqrt(x))
	t_3 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))))
	tmp = 0.0
	if (t_3 <= 1.5)
		tmp = t_2;
	elseif (t_3 <= 2.5)
		tmp = Float64(1.0 + Float64(Float64(t_1 + Float64(z * 0.0)) - sqrt(x)));
	else
		tmp = Float64(t_2 + 2.0);
	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 = t_1 - sqrt(x);
	t_3 = (sqrt((1.0 + z)) - sqrt(z)) + (t_2 + (sqrt((1.0 + y)) - sqrt(y)));
	tmp = 0.0;
	if (t_3 <= 1.5)
		tmp = t_2;
	elseif (t_3 <= 2.5)
		tmp = 1.0 + ((t_1 + (z * 0.0)) - sqrt(x));
	else
		tmp = t_2 + 2.0;
	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[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1.5], t$95$2, If[LessEqual[t$95$3, 2.5], N[(1.0 + N[(N[(t$95$1 + N[(z * 0.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + 2.0), $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))) < 1.5

   1. Initial program 86.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 5.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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 16.6

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 16.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x}} - \sqrt{x} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{1 + x}} - \sqrt{x} \]

       4. sqrt-lowering-sqrt.f64 16.5

          \[\leadsto \sqrt{1 + x} - \color{blue}{\sqrt{x}} \]

   11. Simplified 16.5%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   if 1.5 < (+.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))) < 2.5

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. 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 27.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 27.6

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 27.6%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 30.3

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 30.3%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{z \cdot \left(\sqrt{\frac{1}{z}} + -1 \cdot \sqrt{\frac{1}{z}}\right)}\right) - \sqrt{x}\right) \]
   12. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + z \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \sqrt{\frac{1}{z}}\right)}\right) - \sqrt{x}\right) \]

       2. metadata-eval N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + z \cdot \left(\color{blue}{0} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right) \]

       3. mul0-lft N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + z \cdot \color{blue}{0}\right) - \sqrt{x}\right) \]

       4. *-lowering-*.f64 42.7

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{z \cdot 0}\right) - \sqrt{x}\right) \]

   13. Simplified 42.7%

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{z \cdot 0}\right) - \sqrt{x}\right) \]

   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 99.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   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 49.8

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 49.8

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 49.8%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 47.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 47.2%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 2 + \left(\color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) \]

       6. sqrt-lowering-sqrt.f64 47.2

          \[\leadsto 2 + \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \]

   13. Simplified 47.2%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.5:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;1 + \left(\left(\sqrt{x + 1} + z \cdot 0\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \frac{1}{\sqrt{y} + t\_3}\\ \mathbf{if}\;t\_1 + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, t\_4\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(t\_1 + \left(\left(t\_4 + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (/ 1.0 (+ (sqrt y) t_3))))
   (if (<= (+ t_1 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_3 (sqrt y)))) 0.0001)
     (+ t_2 (+ (fma 0.5 (sqrt (/ 1.0 x)) t_4) t_1))
     (+ t_2 (+ t_1 (- (+ t_4 (fma x 0.5 1.0)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + y));
	double t_4 = 1.0 / (sqrt(y) + t_3);
	double tmp;
	if ((t_1 + ((sqrt((x + 1.0)) - sqrt(x)) + (t_3 - sqrt(y)))) <= 0.0001) {
		tmp = t_2 + (fma(0.5, sqrt((1.0 / x)), t_4) + t_1);
	} else {
		tmp = t_2 + (t_1 + ((t_4 + fma(x, 0.5, 1.0)) - sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(1.0 / Float64(sqrt(y) + t_3))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_3 - sqrt(y)))) <= 0.0001)
		tmp = Float64(t_2 + Float64(fma(0.5, sqrt(Float64(1.0 / x)), t_4) + t_1));
	else
		tmp = Float64(t_2 + Float64(t_1 + Float64(Float64(t_4 + fma(x, 0.5, 1.0)) - sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + 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]), $MachinePrecision], 0.0001], N[(t$95$2 + N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(t$95$1 + N[(N[(t$95$4 + N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00000000000000005e-4

   1. Initial program 52.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 57.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 57.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 89.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 89.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000005e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 53.5

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 53.5%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;t\_2 + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t\_2 + \left(\left(\frac{1}{\sqrt{y} + t\_1} + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (sqrt (+ x 1.0))))
   (if (<= (+ t_2 (+ (- t_3 (sqrt x)) (- t_1 (sqrt y)))) 0.9999999989571969)
     (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) t_3)))
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (+ t_2 (- (+ (/ 1.0 (+ (sqrt y) t_1)) (fma x 0.5 1.0)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if ((t_2 + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)))) <= 0.9999999989571969) {
		tmp = fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + t_3)));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (t_2 + (((1.0 / (sqrt(y) + t_1)) + fma(x, 0.5, 1.0)) - sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y)))) <= 0.9999999989571969)
		tmp = fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + t_3)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(t_2 + Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + fma(x, 0.5, 1.0)) - sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.9999999989571969], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999999998957196934

   1. Initial program 64.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 25.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 48.0

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. div-inv N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}, \frac{1}{\sqrt{x + 1} + \sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   10. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) \]

       8. +-lowering-+.f64 27.3

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) \]

   13. Simplified 27.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

   if 0.999999998957196934 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1\right) + \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 54.6

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 54.6%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 1\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.9999999989571969:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \mathsf{fma}\left(x, 0.5, 1\right)\right) - \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;t\_2 + \left(\left(t\_3 - \sqrt{x}\right) + t\_1\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t\_2 + \left(t\_1 + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (sqrt (+ x 1.0))))
   (if (<= (+ t_2 (+ (- t_3 (sqrt x)) t_1)) 0.0001)
     (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) t_3)))
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (+ t_2 (+ t_1 (fma x 0.5 (- 1.0 (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if ((t_2 + ((t_3 - sqrt(x)) + t_1)) <= 0.0001) {
		tmp = fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + t_3)));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (t_2 + (t_1 + fma(x, 0.5, (1.0 - sqrt(x)))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	t_2 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(t_3 - sqrt(x)) + t_1)) <= 0.0001)
		tmp = fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + t_3)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(t_2 + Float64(t_1 + fma(x, 0.5, Float64(1.0 - sqrt(x))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 0.0001], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$1 + N[(x * 0.5 + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.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))) < 1.00000000000000005e-4

   1. Initial program 52.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 23.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 47.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 47.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. div-inv N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}, \frac{1}{\sqrt{x + 1} + \sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   10. Applied egg-rr 54.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) \]

       8. +-lowering-+.f64 26.9

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) \]

   13. Simplified 26.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

   if 1.00000000000000005e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 53.1

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 53.1%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{1 + y}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \sqrt{z} + \sqrt{1 + z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{t\_1}\right) + \frac{1}{t\_3}\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(t\_2 + \frac{\left(1 + y\right) - y}{t\_1}\right) + \frac{\left(1 + z\right) - z}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt (+ 1.0 y))))
        (t_2 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_3 (+ (sqrt z) (sqrt (+ 1.0 z))))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 1e-5)
     (+ (+ (fma 0.5 (sqrt (/ 1.0 x)) (/ 1.0 t_1)) (/ 1.0 t_3)) t_4)
     (+ t_4 (+ (+ t_2 (/ (- (+ 1.0 y) y) t_1)) (/ (- (+ 1.0 z) z) t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt((1.0 + y));
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = sqrt(z) + sqrt((1.0 + z));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = (fma(0.5, sqrt((1.0 / x)), (1.0 / t_1)) + (1.0 / t_3)) + t_4;
	} else {
		tmp = t_4 + ((t_2 + (((1.0 + y) - y) / t_1)) + (((1.0 + z) - z) / t_3));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(Float64(1.0 + y)))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(sqrt(z) + sqrt(Float64(1.0 + z)))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 1e-5)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / x)), Float64(1.0 / t_1)) + Float64(1.0 / t_3)) + t_4);
	else
		tmp = Float64(t_4 + Float64(Float64(t_2 + Float64(Float64(Float64(1.0 + y) - y) / t_1)) + Float64(Float64(Float64(1.0 + z) - z) / 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[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-5], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], N[(t$95$4 + N[(N[(t$95$2 + N[(N[(N[(1.0 + y), $MachinePrecision] - y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000008e-5

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z}} + \sqrt{z + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \color{blue}{\sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 89.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 89.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. +-lowering-+.f64 98.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 98.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z}} + \sqrt{z + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \color{blue}{\sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 98.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 98.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, t\_1\right) + \frac{1}{\sqrt{z} + t\_2}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), t\_1\right) - \sqrt{x}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

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 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0001)
     (+ (+ (fma 0.5 (sqrt (/ 1.0 x)) t_1) (/ 1.0 (+ (sqrt z) t_2))) t_3)
     (+
      t_3
      (+
       (+ 1.0 (- (fma x (fma x -0.125 0.5) t_1) (sqrt x)))
       (- t_2 (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.0001) {
		tmp = (fma(0.5, sqrt((1.0 / x)), t_1) + (1.0 / (sqrt(z) + t_2))) + t_3;
	} else {
		tmp = t_3 + ((1.0 + (fma(x, fma(x, -0.125, 0.5), t_1) - sqrt(x))) + (t_2 - sqrt(z)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.0001)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / x)), t_1) + Float64(1.0 / Float64(sqrt(z) + t_2))) + t_3);
	else
		tmp = Float64(t_3 + Float64(Float64(1.0 + Float64(fma(x, fma(x, -0.125, 0.5), t_1) - sqrt(x))) + Float64(t_2 - 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[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(t$95$3 + N[(N[(1.0 + N[(N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z}} + \sqrt{z + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \color{blue}{\sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-lowering-+.f64 89.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 89.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. +-lowering-+.f64 98.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 98.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(1 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{8} \cdot x, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right)}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 94.5

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 94.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_1 + \left(t\_2 - \sqrt{y}\right)\right)\right) \leq 2.5:\\ \;\;\;\;1 + \left(t\_2 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + 2\\ \end{array} \end{array} \]

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 y))))
   (if (<=
        (+
         (- (sqrt (+ 1.0 t)) (sqrt t))
         (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ t_1 (- t_2 (sqrt y)))))
        2.5)
     (+ 1.0 (- t_2 (+ (sqrt x) (sqrt y))))
     (+ t_1 2.0))))

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 + y));
	double tmp;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((sqrt((1.0 + z)) - sqrt(z)) + (t_1 + (t_2 - sqrt(y))))) <= 2.5) {
		tmp = 1.0 + (t_2 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_1 + 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    t_2 = sqrt((1.0d0 + y))
    if (((sqrt((1.0d0 + t)) - sqrt(t)) + ((sqrt((1.0d0 + z)) - sqrt(z)) + (t_1 + (t_2 - sqrt(y))))) <= 2.5d0) then
        tmp = 1.0d0 + (t_2 - (sqrt(x) + sqrt(y)))
    else
        tmp = t_1 + 2.0d0
    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)) - Math.sqrt(x);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (t_1 + (t_2 - Math.sqrt(y))))) <= 2.5) {
		tmp = 1.0 + (t_2 - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = t_1 + 2.0;
	}
	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)) - math.sqrt(x)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if ((math.sqrt((1.0 + t)) - math.sqrt(t)) + ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (t_1 + (t_2 - math.sqrt(y))))) <= 2.5:
		tmp = 1.0 + (t_2 - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = t_1 + 2.0
	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 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_1 + Float64(t_2 - sqrt(y))))) <= 2.5)
		tmp = Float64(1.0 + Float64(t_2 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_1 + 2.0);
	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)) - sqrt(x);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (((sqrt((1.0 + t)) - sqrt(t)) + ((sqrt((1.0 + z)) - sqrt(z)) + (t_1 + (t_2 - sqrt(y))))) <= 2.5)
		tmp = 1.0 + (t_2 - (sqrt(x) + sqrt(y)));
	else
		tmp = t_1 + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.5], N[(1.0 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + 2.0), $MachinePrecision]]]]

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))) < 2.5

   1. Initial program 89.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 16.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 16.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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 21.1

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

       9. sqrt-lowering-sqrt.f64 24.3

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   11. Simplified 24.3%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.5 < (+.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 99.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 25.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 25.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 25.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 25.2%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 37.9

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 37.9%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 2 + \left(\color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) \]

       6. sqrt-lowering-sqrt.f64 61.7

          \[\leadsto 2 + \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \]

   13. Simplified 61.7%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.5:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, t\_1\right) + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), t\_1\right) - \sqrt{x}\right)\right) + t\_3\right)\\ \end{array} \end{array} \]

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 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (- (sqrt (+ 1.0 z)) (sqrt z))))
   (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0001)
     (+ t_2 (+ (fma 0.5 (sqrt (/ 1.0 x)) t_1) t_3))
     (+ t_2 (+ (+ 1.0 (- (fma x (fma x -0.125 0.5) t_1) (sqrt x))) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + z)) - sqrt(z);
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.0001) {
		tmp = t_2 + (fma(0.5, sqrt((1.0 / x)), t_1) + t_3);
	} else {
		tmp = t_2 + ((1.0 + (fma(x, fma(x, -0.125, 0.5), t_1) - sqrt(x))) + t_3);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.0001)
		tmp = Float64(t_2 + Float64(fma(0.5, sqrt(Float64(1.0 / x)), t_1) + t_3));
	else
		tmp = Float64(t_2 + Float64(Float64(1.0 + Float64(fma(x, fma(x, -0.125, 0.5), t_1) - sqrt(x))) + 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[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $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[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0001], N[(t$95$2 + N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(1.0 + N[(N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 88.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 96.0

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 96.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   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. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 97.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \color{blue}{\sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 97.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(1 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{8} \cdot x, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right)}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. sqrt-lowering-sqrt.f64 94.5

          \[\leadsto \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 94.5%

      \[\leadsto \left(\color{blue}{\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_1 - \left(\sqrt{x} + -1\right)\right)\right)\\ \end{array} \end{array} \]

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)) (sqrt y))))
   (if (<= t_1 0.0)
     (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))))
     (+
      (- (sqrt (+ 1.0 t)) (sqrt t))
      (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- t_1 (+ (sqrt x) -1.0)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y)) - sqrt(y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + sqrt((x + 1.0)))));
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + ((sqrt((1.0 + z)) - sqrt(z)) + (t_1 - (sqrt(x) + -1.0)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + y)) - sqrt(y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(t_1 - Float64(sqrt(x) + -1.0))));
	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 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.0

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + t} + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \sqrt{1 + t}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{t}\right)}\right) + \sqrt{1 + t} \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \sqrt{t}\right)} + \sqrt{1 + t} \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   5. Simplified 12.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      8. sqrt-lowering-sqrt.f64 38.7

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   8. Simplified 38.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      5. div-inv N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}, \frac{1}{\sqrt{x + 1} + \sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   10. Applied egg-rr 40.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}}\right)} - \left(\sqrt{t} - \sqrt{1 + t}\right) \]

   11. Taylor expanded in t around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) \]

       8. +-lowering-+.f64 25.7

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) \]

   13. Simplified 25.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) + 1\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + 1\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l- N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \left(\sqrt{x} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\left(\sqrt{x} - 1\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. sqrt-lowering-sqrt.f64 44.1

          \[\leadsto \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\color{blue}{\sqrt{x}} - 1\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 44.1%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - 1\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 53.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t\_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.5:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + 2\\ \end{array} \end{array} \]

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 z)) (sqrt z)) (+ t_1 (- (sqrt (+ 1.0 y)) (sqrt y))))
        1.5)
     t_1
     (+ t_1 2.0))))

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 + z)) - sqrt(z)) + (t_1 + (sqrt((1.0 + y)) - sqrt(y)))) <= 1.5) {
		tmp = t_1;
	} else {
		tmp = t_1 + 2.0;
	}
	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)) - sqrt(x)
    if (((sqrt((1.0d0 + z)) - sqrt(z)) + (t_1 + (sqrt((1.0d0 + y)) - sqrt(y)))) <= 1.5d0) then
        tmp = t_1
    else
        tmp = t_1 + 2.0d0
    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)) - Math.sqrt(x);
	double tmp;
	if (((Math.sqrt((1.0 + z)) - Math.sqrt(z)) + (t_1 + (Math.sqrt((1.0 + y)) - Math.sqrt(y)))) <= 1.5) {
		tmp = t_1;
	} else {
		tmp = t_1 + 2.0;
	}
	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)) - math.sqrt(x)
	tmp = 0
	if ((math.sqrt((1.0 + z)) - math.sqrt(z)) + (t_1 + (math.sqrt((1.0 + y)) - math.sqrt(y)))) <= 1.5:
		tmp = t_1
	else:
		tmp = t_1 + 2.0
	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 + z)) - sqrt(z)) + Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)))) <= 1.5)
		tmp = t_1;
	else
		tmp = Float64(t_1 + 2.0);
	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)) - sqrt(x);
	tmp = 0.0;
	if (((sqrt((1.0 + z)) - sqrt(z)) + (t_1 + (sqrt((1.0 + y)) - sqrt(y)))) <= 1.5)
		tmp = t_1;
	else
		tmp = t_1 + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], t$95$1, N[(t$95$1 + 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.5

   1. Initial program 86.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 5.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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 16.6

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 16.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x}} - \sqrt{x} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{1 + x}} - \sqrt{x} \]

       4. sqrt-lowering-sqrt.f64 16.5

          \[\leadsto \sqrt{1 + x} - \color{blue}{\sqrt{x}} \]

   11. Simplified 16.5%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   if 1.5 < (+.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 98.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 32.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 32.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 32.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

   7. Applied egg-rr 32.3%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

      14. sqrt-lowering-sqrt.f64 33.9

          \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

   10. Simplified 33.9%

       \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\left(2 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto 2 + \left(\color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) \]

       6. sqrt-lowering-sqrt.f64 56.5

          \[\leadsto 2 + \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \]

   13. Simplified 56.5%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.5:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.7% 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 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in 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 18.5

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 18.5%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

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 19.8

       \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

8. Simplified 19.8%

   \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
10. Step-by-step derivation

    1. --lowering--.f64 N/A

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{1 + x}} - \sqrt{x} \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \sqrt{\color{blue}{1 + x}} - \sqrt{x} \]

    4. sqrt-lowering-sqrt.f64 15.1

       \[\leadsto \sqrt{1 + x} - \color{blue}{\sqrt{x}} \]

11. Simplified 15.1%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

12. Final simplification 15.1%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]
13. Add Preprocessing

Alternative 21: 34.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \left(\sqrt{x} - \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (- (sqrt x) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (sqrt(x) - 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 = 1.0d0 + (sqrt(x) - 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 1.0 + (Math.sqrt(x) - Math.sqrt(x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (math.sqrt(x) - math.sqrt(x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(sqrt(x) - sqrt(x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (sqrt(x) - 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[(1.0 + N[(N[Sqrt[x], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in 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 18.5

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 18.5%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 18.6

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

7. Applied egg-rr 18.6%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   3. --lowering--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   14. sqrt-lowering-sqrt.f64 36.2

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto 1 + \left(\color{blue}{\sqrt{x}} - \sqrt{x}\right) \]
12. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 35.1

       \[\leadsto 1 + \left(\color{blue}{\sqrt{x}} - \sqrt{x}\right) \]

13. Simplified 35.1%

    \[\leadsto 1 + \left(\color{blue}{\sqrt{x}} - \sqrt{x}\right) \]
14. Add Preprocessing

Alternative 22: 10.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \sqrt{z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (sqrt z)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + 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 = 1.0d0 + 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 1.0 + Math.sqrt(z);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + math.sqrt(z)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + sqrt(z))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + 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[(1.0 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in 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 18.5

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 18.5%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 18.6

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

7. Applied egg-rr 18.6%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)\right) - \sqrt{x}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   3. --lowering--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right)} - \sqrt{x}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)}\right) - \sqrt{x}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \color{blue}{\frac{z}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) - \sqrt{x}\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) - \sqrt{x}\right) \]

   14. sqrt-lowering-sqrt.f64 36.2

       \[\leadsto 1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + z} + \frac{z}{\sqrt{y} - \sqrt{z}}\right)\right) - \sqrt{x}\right)} \]

11. Taylor expanded in z around inf

    \[\leadsto 1 + \color{blue}{\sqrt{z}} \]
12. Step-by-step derivation

    1. sqrt-lowering-sqrt.f64 17.0

       \[\leadsto 1 + \color{blue}{\sqrt{z}} \]

13. Simplified 17.0%

    \[\leadsto 1 + \color{blue}{\sqrt{z}} \]
14. Add Preprocessing

Alternative 23: 7.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(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 = sqrt(y)
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(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(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[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in 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 18.5

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 18.5%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right)} \cdot \frac{1}{\sqrt{y} - \sqrt{z}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \color{blue}{\frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\color{blue}{\sqrt{y}} - \sqrt{z}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 18.6

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(y - z\right) \cdot \frac{1}{\sqrt{y} - \color{blue}{\sqrt{z}}}\right)\right) \]

7. Applied egg-rr 18.6%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(y - z\right) \cdot \frac{1}{\sqrt{y} - \sqrt{z}}}\right)\right) \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\sqrt{y}} \]
9. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 6.9

      \[\leadsto \color{blue}{\sqrt{y}} \]

10. Simplified 6.9%

    \[\leadsto \color{blue}{\sqrt{y}} \]
11. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024199 
(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))))