Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 69.4% → 99.9%

Time: 9.9s

Alternatives: 7

Speedup: 1.2×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(y + \frac{x - z}{y} \cdot \left(x + z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ y (* (/ (- x z) y) (+ x z)))))

C

double code(double x, double y, double z) {
	return 0.5 * (y + (((x - z) / y) * (x + z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (y + (((x - z) / y) * (x + z)))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (y + (((x - z) / y) * (x + z)));
}

Python

def code(x, y, z):
	return 0.5 * (y + (((x - z) / y) * (x + z)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(y + Float64(Float64(Float64(x - z) / y) * Float64(x + z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (y + (((x - z) / y) * (x + z)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(y + N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. remove-double-neg 72.9%

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

   2. distribute-lft-neg-out 72.9%

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

   3. distribute-frac-neg2 72.9%

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

   4. distribute-frac-neg 72.9%

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

   5. neg-mul-1 72.9%

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

   6. distribute-lft-neg-out 72.9%

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

   7. *-commutative 72.9%

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

   8. distribute-lft-neg-in 72.9%

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

   9. times-frac 72.9%

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

   10. metadata-eval 72.9%

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

   11. metadata-eval 72.9%

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

   12. associate--l+ 72.9%

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

   13. fma-define 74.8%

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

3. Simplified 74.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 81.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. associate--l+ 81.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

   2. div-sub 84.6%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

7. Simplified 84.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
8. Step-by-step derivation

   1. unpow2 84.6%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   2. pow2 84.6%

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

   3. difference-of-squares 88.2%

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

9. Applied egg-rr 88.2%

   \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
10. Step-by-step derivation

    1. associate-/l* 99.9%

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

    2. *-commutative 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \frac{x - z}{y} \cdot \left(x + z\right)\right) \]
13. Add Preprocessing

Alternative 2: 45.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{if}\;z \leq 1.05 \cdot 10^{-244}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 0.000255:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 650000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* 0.5 (/ x y)))))
   (if (<= z 1.05e-244)
     (* 0.5 y)
     (if (<= z 8.8e-190)
       t_0
       (if (<= z 7.6e-136)
         (* 0.5 y)
         (if (<= z 1.7e-105)
           t_0
           (if (<= z 2.6e-46)
             (* 0.5 y)
             (if (<= z 0.000255)
               t_0
               (if (<= z 650000000.0)
                 (* 0.5 y)
                 (if (<= z 4.5e+55) t_0 (* z (* z (/ -0.5 y)))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (0.5 * (x / y));
	double tmp;
	if (z <= 1.05e-244) {
		tmp = 0.5 * y;
	} else if (z <= 8.8e-190) {
		tmp = t_0;
	} else if (z <= 7.6e-136) {
		tmp = 0.5 * y;
	} else if (z <= 1.7e-105) {
		tmp = t_0;
	} else if (z <= 2.6e-46) {
		tmp = 0.5 * y;
	} else if (z <= 0.000255) {
		tmp = t_0;
	} else if (z <= 650000000.0) {
		tmp = 0.5 * y;
	} else if (z <= 4.5e+55) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.5d0 * (x / y))
    if (z <= 1.05d-244) then
        tmp = 0.5d0 * y
    else if (z <= 8.8d-190) then
        tmp = t_0
    else if (z <= 7.6d-136) then
        tmp = 0.5d0 * y
    else if (z <= 1.7d-105) then
        tmp = t_0
    else if (z <= 2.6d-46) then
        tmp = 0.5d0 * y
    else if (z <= 0.000255d0) then
        tmp = t_0
    else if (z <= 650000000.0d0) then
        tmp = 0.5d0 * y
    else if (z <= 4.5d+55) then
        tmp = t_0
    else
        tmp = z * (z * ((-0.5d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (0.5 * (x / y));
	double tmp;
	if (z <= 1.05e-244) {
		tmp = 0.5 * y;
	} else if (z <= 8.8e-190) {
		tmp = t_0;
	} else if (z <= 7.6e-136) {
		tmp = 0.5 * y;
	} else if (z <= 1.7e-105) {
		tmp = t_0;
	} else if (z <= 2.6e-46) {
		tmp = 0.5 * y;
	} else if (z <= 0.000255) {
		tmp = t_0;
	} else if (z <= 650000000.0) {
		tmp = 0.5 * y;
	} else if (z <= 4.5e+55) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (0.5 * (x / y))
	tmp = 0
	if z <= 1.05e-244:
		tmp = 0.5 * y
	elif z <= 8.8e-190:
		tmp = t_0
	elif z <= 7.6e-136:
		tmp = 0.5 * y
	elif z <= 1.7e-105:
		tmp = t_0
	elif z <= 2.6e-46:
		tmp = 0.5 * y
	elif z <= 0.000255:
		tmp = t_0
	elif z <= 650000000.0:
		tmp = 0.5 * y
	elif z <= 4.5e+55:
		tmp = t_0
	else:
		tmp = z * (z * (-0.5 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(0.5 * Float64(x / y)))
	tmp = 0.0
	if (z <= 1.05e-244)
		tmp = Float64(0.5 * y);
	elseif (z <= 8.8e-190)
		tmp = t_0;
	elseif (z <= 7.6e-136)
		tmp = Float64(0.5 * y);
	elseif (z <= 1.7e-105)
		tmp = t_0;
	elseif (z <= 2.6e-46)
		tmp = Float64(0.5 * y);
	elseif (z <= 0.000255)
		tmp = t_0;
	elseif (z <= 650000000.0)
		tmp = Float64(0.5 * y);
	elseif (z <= 4.5e+55)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (0.5 * (x / y));
	tmp = 0.0;
	if (z <= 1.05e-244)
		tmp = 0.5 * y;
	elseif (z <= 8.8e-190)
		tmp = t_0;
	elseif (z <= 7.6e-136)
		tmp = 0.5 * y;
	elseif (z <= 1.7e-105)
		tmp = t_0;
	elseif (z <= 2.6e-46)
		tmp = 0.5 * y;
	elseif (z <= 0.000255)
		tmp = t_0;
	elseif (z <= 650000000.0)
		tmp = 0.5 * y;
	elseif (z <= 4.5e+55)
		tmp = t_0;
	else
		tmp = z * (z * (-0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.05e-244], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 8.8e-190], t$95$0, If[LessEqual[z, 7.6e-136], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 1.7e-105], t$95$0, If[LessEqual[z, 2.6e-46], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 0.000255], t$95$0, If[LessEqual[z, 650000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 4.5e+55], t$95$0, N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.05000000000000001e-244 or 8.80000000000000017e-190 < z < 7.6000000000000005e-136 or 1.69999999999999996e-105 < z < 2.6000000000000002e-46 or 2.55e-4 < z < 6.5e8

   1. Initial program 72.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 41.2%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 41.2%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 1.05000000000000001e-244 < z < 8.80000000000000017e-190 or 7.6000000000000005e-136 < z < 1.69999999999999996e-105 or 2.6000000000000002e-46 < z < 2.55e-4 or 6.5e8 < z < 4.49999999999999998e55

   1. Initial program 82.6%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

      3. associate-/l* 82.4%

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

      4. add-sqr-sqrt 82.4%

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

      5. pow2 82.4%

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

      6. hypot-define 82.4%

         \[\leadsto {\left(y \cdot \frac{2}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}\right)}^{-1} \]

      7. pow2 82.4%

         \[\leadsto {\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}\right)}^{-1} \]

   4. Applied egg-rr 82.4%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]

   5. Taylor expanded in x around inf 63.8%

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

      1. associate-*r/ 63.8%

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

      2. unpow2 63.8%

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

      3. associate-/r* 71.3%

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

   7. Applied egg-rr 71.3%

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

      1. unpow-1 71.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot 2}{x}}{x}}} \]

      2. clear-num 71.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 2}{x}}} \]

      3. associate-/r/ 71.2%

         \[\leadsto \color{blue}{\frac{x}{y \cdot 2} \cdot x} \]

      4. *-un-lft-identity 71.2%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot 2} \cdot x \]

      5. *-commutative 71.2%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{2 \cdot y}} \cdot x \]

      6. times-frac 71.2%

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

      7. metadata-eval 71.2%

         \[\leadsto \left(\color{blue}{0.5} \cdot \frac{x}{y}\right) \cdot x \]

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{y}\right) \cdot x} \]

   if 4.49999999999999998e55 < z

   1. Initial program 63.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 65.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot {z}^{2}}{y}} \]

      2. metadata-eval 65.0%

         \[\leadsto \frac{\color{blue}{\left(-0.5\right)} \cdot {z}^{2}}{y} \]

      3. distribute-lft-neg-in 65.0%

         \[\leadsto \frac{\color{blue}{-0.5 \cdot {z}^{2}}}{y} \]

      4. *-commutative 65.0%

         \[\leadsto \frac{-\color{blue}{{z}^{2} \cdot 0.5}}{y} \]

      5. distribute-neg-frac 65.0%

         \[\leadsto \color{blue}{-\frac{{z}^{2} \cdot 0.5}{y}} \]

      6. associate-*r/ 65.0%

         \[\leadsto -\color{blue}{{z}^{2} \cdot \frac{0.5}{y}} \]

      7. distribute-rgt-neg-in 65.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot \left(-\frac{0.5}{y}\right)} \]

      8. distribute-neg-frac 65.0%

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{-0.5}{y}} \]

      9. metadata-eval 65.0%

         \[\leadsto {z}^{2} \cdot \frac{\color{blue}{-0.5}}{y} \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{{z}^{2} \cdot \frac{-0.5}{y}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 55.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left({z}^{2} \cdot \frac{-0.5}{y}\right) \cdot \left({z}^{2} \cdot \frac{-0.5}{y}\right)\right) \cdot \left({z}^{2} \cdot \frac{-0.5}{y}\right)}} \]

      2. pow3 55.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({z}^{2} \cdot \frac{-0.5}{y}\right)}^{3}}} \]

   7. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({z}^{2} \cdot \frac{-0.5}{y}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 65.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{-0.5}{y}} \]

      2. *-commutative 65.0%

         \[\leadsto \color{blue}{\frac{-0.5}{y} \cdot {z}^{2}} \]

      3. unpow2 65.0%

         \[\leadsto \frac{-0.5}{y} \cdot \color{blue}{\left(z \cdot z\right)} \]

      4. associate-*r* 65.0%

         \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-244}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 0.000255:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 650000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{if}\;z \leq 9.2 \cdot 10^{-245}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{x}{\frac{y \cdot -2}{-x}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 0.000365:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* 0.5 (/ x y)))))
   (if (<= z 9.2e-245)
     (* 0.5 y)
     (if (<= z 8.8e-190)
       (/ x (/ (* y -2.0) (- x)))
       (if (<= z 7e-136)
         (* 0.5 y)
         (if (<= z 1.26e-105)
           t_0
           (if (<= z 2.2e-46)
             (* 0.5 y)
             (if (<= z 0.000365)
               t_0
               (if (<= z 620000000.0)
                 (* 0.5 y)
                 (if (<= z 6.5e+54) t_0 (* z (* z (/ -0.5 y)))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (0.5 * (x / y));
	double tmp;
	if (z <= 9.2e-245) {
		tmp = 0.5 * y;
	} else if (z <= 8.8e-190) {
		tmp = x / ((y * -2.0) / -x);
	} else if (z <= 7e-136) {
		tmp = 0.5 * y;
	} else if (z <= 1.26e-105) {
		tmp = t_0;
	} else if (z <= 2.2e-46) {
		tmp = 0.5 * y;
	} else if (z <= 0.000365) {
		tmp = t_0;
	} else if (z <= 620000000.0) {
		tmp = 0.5 * y;
	} else if (z <= 6.5e+54) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.5d0 * (x / y))
    if (z <= 9.2d-245) then
        tmp = 0.5d0 * y
    else if (z <= 8.8d-190) then
        tmp = x / ((y * (-2.0d0)) / -x)
    else if (z <= 7d-136) then
        tmp = 0.5d0 * y
    else if (z <= 1.26d-105) then
        tmp = t_0
    else if (z <= 2.2d-46) then
        tmp = 0.5d0 * y
    else if (z <= 0.000365d0) then
        tmp = t_0
    else if (z <= 620000000.0d0) then
        tmp = 0.5d0 * y
    else if (z <= 6.5d+54) then
        tmp = t_0
    else
        tmp = z * (z * ((-0.5d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (0.5 * (x / y));
	double tmp;
	if (z <= 9.2e-245) {
		tmp = 0.5 * y;
	} else if (z <= 8.8e-190) {
		tmp = x / ((y * -2.0) / -x);
	} else if (z <= 7e-136) {
		tmp = 0.5 * y;
	} else if (z <= 1.26e-105) {
		tmp = t_0;
	} else if (z <= 2.2e-46) {
		tmp = 0.5 * y;
	} else if (z <= 0.000365) {
		tmp = t_0;
	} else if (z <= 620000000.0) {
		tmp = 0.5 * y;
	} else if (z <= 6.5e+54) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (0.5 * (x / y))
	tmp = 0
	if z <= 9.2e-245:
		tmp = 0.5 * y
	elif z <= 8.8e-190:
		tmp = x / ((y * -2.0) / -x)
	elif z <= 7e-136:
		tmp = 0.5 * y
	elif z <= 1.26e-105:
		tmp = t_0
	elif z <= 2.2e-46:
		tmp = 0.5 * y
	elif z <= 0.000365:
		tmp = t_0
	elif z <= 620000000.0:
		tmp = 0.5 * y
	elif z <= 6.5e+54:
		tmp = t_0
	else:
		tmp = z * (z * (-0.5 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(0.5 * Float64(x / y)))
	tmp = 0.0
	if (z <= 9.2e-245)
		tmp = Float64(0.5 * y);
	elseif (z <= 8.8e-190)
		tmp = Float64(x / Float64(Float64(y * -2.0) / Float64(-x)));
	elseif (z <= 7e-136)
		tmp = Float64(0.5 * y);
	elseif (z <= 1.26e-105)
		tmp = t_0;
	elseif (z <= 2.2e-46)
		tmp = Float64(0.5 * y);
	elseif (z <= 0.000365)
		tmp = t_0;
	elseif (z <= 620000000.0)
		tmp = Float64(0.5 * y);
	elseif (z <= 6.5e+54)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (0.5 * (x / y));
	tmp = 0.0;
	if (z <= 9.2e-245)
		tmp = 0.5 * y;
	elseif (z <= 8.8e-190)
		tmp = x / ((y * -2.0) / -x);
	elseif (z <= 7e-136)
		tmp = 0.5 * y;
	elseif (z <= 1.26e-105)
		tmp = t_0;
	elseif (z <= 2.2e-46)
		tmp = 0.5 * y;
	elseif (z <= 0.000365)
		tmp = t_0;
	elseif (z <= 620000000.0)
		tmp = 0.5 * y;
	elseif (z <= 6.5e+54)
		tmp = t_0;
	else
		tmp = z * (z * (-0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 9.2e-245], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 8.8e-190], N[(x / N[(N[(y * -2.0), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e-136], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 1.26e-105], t$95$0, If[LessEqual[z, 2.2e-46], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 0.000365], t$95$0, If[LessEqual[z, 620000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 6.5e+54], t$95$0, N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 9.2000000000000007e-245 or 8.80000000000000017e-190 < z < 7.00000000000000058e-136 or 1.2600000000000001e-105 < z < 2.2000000000000001e-46 or 3.6499999999999998e-4 < z < 6.2e8

   1. Initial program 72.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 41.2%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 41.2%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 9.2000000000000007e-245 < z < 8.80000000000000017e-190

   1. Initial program 74.6%

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

      1. clear-num 74.4%

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

      2. inv-pow 74.4%

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

      3. associate-/l* 74.4%

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

      4. add-sqr-sqrt 74.4%

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

      5. pow2 74.4%

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

      6. hypot-define 74.4%

         \[\leadsto {\left(y \cdot \frac{2}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}\right)}^{-1} \]

      7. pow2 74.4%

         \[\leadsto {\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}\right)}^{-1} \]

   4. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]

   5. Taylor expanded in x around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. unpow2 64.3%

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

      3. associate-/r* 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. unpow-1 73.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot 2}{x}}{x}}} \]

      2. clear-num 73.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 2}{x}}} \]

      3. frac-2neg 73.3%

         \[\leadsto \color{blue}{\frac{-x}{-\frac{y \cdot 2}{x}}} \]

      4. distribute-neg-frac 73.3%

         \[\leadsto \frac{-x}{\color{blue}{\frac{-y \cdot 2}{x}}} \]

      5. distribute-rgt-neg-in 73.3%

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

      6. metadata-eval 73.3%

         \[\leadsto \frac{-x}{\frac{y \cdot \color{blue}{-2}}{x}} \]

   9. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{-x}{\frac{y \cdot -2}{x}}} \]

   if 7.00000000000000058e-136 < z < 1.2600000000000001e-105 or 2.2000000000000001e-46 < z < 3.6499999999999998e-4 or 6.2e8 < z < 6.5e54

   1. Initial program 86.0%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

      3. associate-/l* 85.7%

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

      4. add-sqr-sqrt 85.7%

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

      5. pow2 85.7%

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

      6. hypot-define 85.7%

         \[\leadsto {\left(y \cdot \frac{2}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}\right)}^{-1} \]

      7. pow2 85.7%

         \[\leadsto {\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}\right)}^{-1} \]

   4. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]

   5. Taylor expanded in x around inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. unpow2 63.6%

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

      3. associate-/r* 70.5%

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

   7. Applied egg-rr 70.5%

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

      1. unpow-1 70.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot 2}{x}}{x}}} \]

      2. clear-num 70.5%

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 2}{x}}} \]

      3. associate-/r/ 70.4%

         \[\leadsto \color{blue}{\frac{x}{y \cdot 2} \cdot x} \]

      4. *-un-lft-identity 70.4%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot 2} \cdot x \]

      5. *-commutative 70.4%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{2 \cdot y}} \cdot x \]

      6. times-frac 70.4%

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

      7. metadata-eval 70.4%

         \[\leadsto \left(\color{blue}{0.5} \cdot \frac{x}{y}\right) \cdot x \]

   9. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{y}\right) \cdot x} \]

   if 6.5e54 < z

   1. Initial program 63.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 65.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot {z}^{2}}{y}} \]

      2. metadata-eval 65.0%

         \[\leadsto \frac{\color{blue}{\left(-0.5\right)} \cdot {z}^{2}}{y} \]

      3. distribute-lft-neg-in 65.0%

         \[\leadsto \frac{\color{blue}{-0.5 \cdot {z}^{2}}}{y} \]

      4. *-commutative 65.0%

         \[\leadsto \frac{-\color{blue}{{z}^{2} \cdot 0.5}}{y} \]

      5. distribute-neg-frac 65.0%

         \[\leadsto \color{blue}{-\frac{{z}^{2} \cdot 0.5}{y}} \]

      6. associate-*r/ 65.0%

         \[\leadsto -\color{blue}{{z}^{2} \cdot \frac{0.5}{y}} \]

      7. distribute-rgt-neg-in 65.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot \left(-\frac{0.5}{y}\right)} \]

      8. distribute-neg-frac 65.0%

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{-0.5}{y}} \]

      9. metadata-eval 65.0%

         \[\leadsto {z}^{2} \cdot \frac{\color{blue}{-0.5}}{y} \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{{z}^{2} \cdot \frac{-0.5}{y}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 55.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left({z}^{2} \cdot \frac{-0.5}{y}\right) \cdot \left({z}^{2} \cdot \frac{-0.5}{y}\right)\right) \cdot \left({z}^{2} \cdot \frac{-0.5}{y}\right)}} \]

      2. pow3 55.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({z}^{2} \cdot \frac{-0.5}{y}\right)}^{3}}} \]

   7. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({z}^{2} \cdot \frac{-0.5}{y}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 65.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{-0.5}{y}} \]

      2. *-commutative 65.0%

         \[\leadsto \color{blue}{\frac{-0.5}{y} \cdot {z}^{2}} \]

      3. unpow2 65.0%

         \[\leadsto \frac{-0.5}{y} \cdot \color{blue}{\left(z \cdot z\right)} \]

      4. associate-*r* 65.0%

         \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-245}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{x}{\frac{y \cdot -2}{-x}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{-105}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 0.000365:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+134) (* 0.5 (+ y (* (- x z) (/ z y)))) (* x (* 0.5 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+134) {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	} else {
		tmp = x * (0.5 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5d+134) then
        tmp = 0.5d0 * (y + ((x - z) * (z / y)))
    else
        tmp = x * (0.5d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+134) {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	} else {
		tmp = x * (0.5 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5e+134:
		tmp = 0.5 * (y + ((x - z) * (z / y)))
	else:
		tmp = x * (0.5 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+134)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(z / y))));
	else
		tmp = Float64(x * Float64(0.5 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5e+134)
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	else
		tmp = x * (0.5 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5e+134], N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999981e134

   1. Initial program 74.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. remove-double-neg 74.3%

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

      2. distribute-lft-neg-out 74.3%

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

      3. distribute-frac-neg2 74.3%

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

      4. distribute-frac-neg 74.3%

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

      5. neg-mul-1 74.3%

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

      6. distribute-lft-neg-out 74.3%

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

      7. *-commutative 74.3%

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

      8. distribute-lft-neg-in 74.3%

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

      9. times-frac 74.3%

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

      10. metadata-eval 74.3%

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

      11. metadata-eval 74.3%

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

      12. associate--l+ 74.3%

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

      13. fma-define 75.7%

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

   3. Simplified 75.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 84.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 84.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 87.5%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 87.5%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. pow2 87.5%

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

      3. difference-of-squares 89.7%

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

   9. Applied egg-rr 89.7%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. div-inv 89.7%

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

       2. *-commutative 89.7%

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

       3. associate-*l* 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around 0 73.7%

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

   if 4.99999999999999981e134 < x

   1. Initial program 63.2%

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

      1. clear-num 63.2%

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

      2. inv-pow 63.2%

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

      3. associate-/l* 63.3%

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

      4. add-sqr-sqrt 63.3%

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

      5. pow2 63.3%

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

      6. hypot-define 63.3%

         \[\leadsto {\left(y \cdot \frac{2}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}\right)}^{-1} \]

      7. pow2 63.3%

         \[\leadsto {\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}\right)}^{-1} \]

   4. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]

   5. Taylor expanded in x around inf 63.7%

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

      1. associate-*r/ 63.6%

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

      2. unpow2 63.6%

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

      3. associate-/r* 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. unpow-1 71.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot 2}{x}}{x}}} \]

      2. clear-num 71.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 2}{x}}} \]

      3. associate-/r/ 71.8%

         \[\leadsto \color{blue}{\frac{x}{y \cdot 2} \cdot x} \]

      4. *-un-lft-identity 71.8%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot 2} \cdot x \]

      5. *-commutative 71.8%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{2 \cdot y}} \cdot x \]

      6. times-frac 71.8%

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

      7. metadata-eval 71.8%

         \[\leadsto \left(\color{blue}{0.5} \cdot \frac{x}{y}\right) \cdot x \]

   9. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{y}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x + z\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 4.6e+50)
   (* 0.5 (+ y (* (+ x z) (/ x y))))
   (* 0.5 (+ y (* (- x z) (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.6e+50) {
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	} else {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 4.6d+50) then
        tmp = 0.5d0 * (y + ((x + z) * (x / y)))
    else
        tmp = 0.5d0 * (y + ((x - z) * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.6e+50) {
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	} else {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.6e+50:
		tmp = 0.5 * (y + ((x + z) * (x / y)))
	else:
		tmp = 0.5 * (y + ((x - z) * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.6e+50)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x + z) * Float64(x / y))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 4.6e+50)
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	else
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.6e+50], N[(0.5 * N[(y + N[(N[(x + z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.59999999999999994e50

   1. Initial program 74.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. remove-double-neg 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-frac-neg2 74.4%

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

      4. distribute-frac-neg 74.4%

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

      5. neg-mul-1 74.4%

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

      6. distribute-lft-neg-out 74.4%

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

      7. *-commutative 74.4%

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

      8. distribute-lft-neg-in 74.4%

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

      9. times-frac 74.4%

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

      10. metadata-eval 74.4%

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

      11. metadata-eval 74.4%

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

      12. associate--l+ 74.4%

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

      13. fma-define 75.8%

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

   3. Simplified 75.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 84.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 84.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 87.3%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 87.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 87.3%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. pow2 87.3%

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

      3. difference-of-squares 89.6%

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

   9. Applied egg-rr 89.6%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 99.9%

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

       2. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around inf 79.9%

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

   if 4.59999999999999994e50 < z

   1. Initial program 64.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. remove-double-neg 64.0%

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

      2. distribute-lft-neg-out 64.0%

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

      3. distribute-frac-neg2 64.0%

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

      4. distribute-frac-neg 64.0%

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

      5. neg-mul-1 64.0%

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

      6. distribute-lft-neg-out 64.0%

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

      7. *-commutative 64.0%

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

      8. distribute-lft-neg-in 64.0%

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

      9. times-frac 64.0%

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

      10. metadata-eval 64.0%

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

      11. metadata-eval 64.0%

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

      12. associate--l+ 64.0%

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

      13. fma-define 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 63.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 69.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 69.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 69.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. pow2 69.1%

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

      3. difference-of-squares 79.9%

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

   9. Applied egg-rr 79.9%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. div-inv 79.8%

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

       2. *-commutative 79.8%

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

       3. associate-*l* 100.0%

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

   11. Applied egg-rr 100.0%

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

   12. Taylor expanded in x around 0 94.7%

       \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x + z\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1200000000000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1200000000000.0) (* x (* 0.5 (/ x y))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1200000000000.0) {
		tmp = x * (0.5 * (x / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1200000000000.0d0) then
        tmp = x * (0.5d0 * (x / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1200000000000.0) {
		tmp = x * (0.5 * (x / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1200000000000.0:
		tmp = x * (0.5 * (x / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1200000000000.0)
		tmp = Float64(x * Float64(0.5 * Float64(x / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1200000000000.0)
		tmp = x * (0.5 * (x / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1200000000000.0], N[(x * N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e12

   1. Initial program 78.6%

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

      1. clear-num 78.5%

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

      2. inv-pow 78.5%

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

      3. associate-/l* 78.5%

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

      4. add-sqr-sqrt 78.5%

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

      5. pow2 78.5%

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

      6. hypot-define 78.5%

         \[\leadsto {\left(y \cdot \frac{2}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}\right)}^{-1} \]

      7. pow2 78.5%

         \[\leadsto {\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}\right)}^{-1} \]

   4. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]

   5. Taylor expanded in x around inf 35.9%

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

      1. associate-*r/ 35.9%

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

      2. unpow2 35.9%

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

      3. associate-/r* 40.2%

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

   7. Applied egg-rr 40.2%

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

      1. unpow-1 40.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot 2}{x}}{x}}} \]

      2. clear-num 40.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 2}{x}}} \]

      3. associate-/r/ 40.2%

         \[\leadsto \color{blue}{\frac{x}{y \cdot 2} \cdot x} \]

      4. *-un-lft-identity 40.2%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot 2} \cdot x \]

      5. *-commutative 40.2%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{2 \cdot y}} \cdot x \]

      6. times-frac 40.2%

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

      7. metadata-eval 40.2%

         \[\leadsto \left(\color{blue}{0.5} \cdot \frac{x}{y}\right) \cdot x \]

   9. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{y}\right) \cdot x} \]

   if 1.2e12 < y

   1. Initial program 54.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.7%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 62.7%

      \[\leadsto \color{blue}{y \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1200000000000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

double code(double x, double y, double z) {
	return 0.5 * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

def code(x, y, z):
	return 0.5 * y

Julia

function code(x, y, z)
	return Float64(0.5 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * y;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 36.5%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
4. Step-by-step derivation

   1. *-commutative 36.5%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

5. Simplified 36.5%

   \[\leadsto \color{blue}{y \cdot 0.5} \]

6. Final simplification 36.5%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024053 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))