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

Percentage Accurate: 68.4% → 99.9%

Time: 15.2s

Alternatives: 12

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: 68.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. add-sqr-sqrt 71.1%

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

   2. difference-of-squares 73.5%

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

   3. hypot-def 74.7%

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

   4. hypot-def 75.9%

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

4. Applied egg-rr 75.9%

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

5. Taylor expanded in y around 0 63.5%

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

   1. distribute-lft-out 63.5%

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

   2. distribute-lft-out 63.5%

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

   3. +-commutative 63.5%

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

   4. associate-/l* 69.7%

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

   5. +-commutative 69.7%

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

7. Simplified 69.7%

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

8. Taylor expanded in y around 0 89.9%

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

   1. associate-/l* 99.9%

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

   2. +-commutative 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 48.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{if}\;y \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ (+ z x) y)))))
   (if (<= y 5.8e-200)
     t_0
     (if (<= y 4e-145)
       (* x (* x (/ 0.5 y)))
       (if (<= y 2.4e-44)
         t_0
         (if (<= y 3e+94) (* (/ x y) (/ x 2.0)) (* 0.5 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * ((z + x) / y));
	double tmp;
	if (y <= 5.8e-200) {
		tmp = t_0;
	} else if (y <= 4e-145) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 2.4e-44) {
		tmp = t_0;
	} else if (y <= 3e+94) {
		tmp = (x / y) * (x / 2.0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * ((z + x) / y))
    if (y <= 5.8d-200) then
        tmp = t_0
    else if (y <= 4d-145) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 2.4d-44) then
        tmp = t_0
    else if (y <= 3d+94) then
        tmp = (x / y) * (x / 2.0d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * ((z + x) / y));
	double tmp;
	if (y <= 5.8e-200) {
		tmp = t_0;
	} else if (y <= 4e-145) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 2.4e-44) {
		tmp = t_0;
	} else if (y <= 3e+94) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * ((z + x) / y))
	tmp = 0
	if y <= 5.8e-200:
		tmp = t_0
	elif y <= 4e-145:
		tmp = x * (x * (0.5 / y))
	elif y <= 2.4e-44:
		tmp = t_0
	elif y <= 3e+94:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(Float64(z + x) / y)))
	tmp = 0.0
	if (y <= 5.8e-200)
		tmp = t_0;
	elseif (y <= 4e-145)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 2.4e-44)
		tmp = t_0;
	elseif (y <= 3e+94)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * ((z + x) / y));
	tmp = 0.0;
	if (y <= 5.8e-200)
		tmp = t_0;
	elseif (y <= 4e-145)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 2.4e-44)
		tmp = t_0;
	elseif (y <= 3e+94)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.8e-200], t$95$0, If[LessEqual[y, 4e-145], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-44], t$95$0, If[LessEqual[y, 3e+94], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.8e-200 or 3.99999999999999966e-145 < y < 2.40000000000000009e-44

   1. Initial program 76.8%

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

      1. add-sqr-sqrt 76.8%

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

      2. difference-of-squares 79.6%

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

      3. hypot-def 80.8%

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

      4. hypot-def 82.0%

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

   4. Applied egg-rr 82.0%

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

   5. Taylor expanded in y around 0 66.9%

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

      1. associate-/l* 69.7%

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

      2. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in x around 0 40.3%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   10. Simplified 40.3%

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

   11. Taylor expanded in y around 0 39.3%

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

       1. +-commutative 39.3%

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

       2. associate-*r/ 42.0%

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

   13. Simplified 42.0%

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

   if 5.8e-200 < y < 3.99999999999999966e-145

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 43.1%

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

      1. div-inv 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-/r* 43.1%

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

      4. metadata-eval 43.1%

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

      5. unpow2 43.1%

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

      6. associate-*l* 43.1%

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

   5. Applied egg-rr 43.1%

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

   if 2.40000000000000009e-44 < y < 3.0000000000000001e94

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. unpow2 35.6%

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

      2. times-frac 44.4%

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

   5. Applied egg-rr 44.4%

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

   if 3.0000000000000001e94 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-44}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-199}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z + x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e-199)
   (* -0.5 (/ z (/ y (+ z x))))
   (if (<= y 1.1e-146)
     (* x (* x (/ 0.5 y)))
     (if (<= y 3.2e-47)
       (* -0.5 (* z (/ (+ z x) y)))
       (if (<= y 4e+93) (* (/ x y) (/ x 2.0)) (* 0.5 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e-199) {
		tmp = -0.5 * (z / (y / (z + x)));
	} else if (y <= 1.1e-146) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3.2e-47) {
		tmp = -0.5 * (z * ((z + x) / y));
	} else if (y <= 4e+93) {
		tmp = (x / y) * (x / 2.0);
	} 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 <= 1.05d-199) then
        tmp = (-0.5d0) * (z / (y / (z + x)))
    else if (y <= 1.1d-146) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 3.2d-47) then
        tmp = (-0.5d0) * (z * ((z + x) / y))
    else if (y <= 4d+93) then
        tmp = (x / y) * (x / 2.0d0)
    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 <= 1.05e-199) {
		tmp = -0.5 * (z / (y / (z + x)));
	} else if (y <= 1.1e-146) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3.2e-47) {
		tmp = -0.5 * (z * ((z + x) / y));
	} else if (y <= 4e+93) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.05e-199:
		tmp = -0.5 * (z / (y / (z + x)))
	elif y <= 1.1e-146:
		tmp = x * (x * (0.5 / y))
	elif y <= 3.2e-47:
		tmp = -0.5 * (z * ((z + x) / y))
	elif y <= 4e+93:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.05e-199)
		tmp = Float64(-0.5 * Float64(z / Float64(y / Float64(z + x))));
	elseif (y <= 1.1e-146)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 3.2e-47)
		tmp = Float64(-0.5 * Float64(z * Float64(Float64(z + x) / y)));
	elseif (y <= 4e+93)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.05e-199)
		tmp = -0.5 * (z / (y / (z + x)));
	elseif (y <= 1.1e-146)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 3.2e-47)
		tmp = -0.5 * (z * ((z + x) / y));
	elseif (y <= 4e+93)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.05e-199], N[(-0.5 * N[(z / N[(y / N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-146], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-47], N[(-0.5 * N[(z * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+93], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.05000000000000001e-199

   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. add-sqr-sqrt 74.6%

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

      2. difference-of-squares 77.0%

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

      3. hypot-def 78.3%

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

      4. hypot-def 79.7%

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

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in y around 0 64.6%

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

      1. associate-/l* 67.8%

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

      2. +-commutative 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around 0 39.2%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 39.2%

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

      2. neg-mul-1 39.2%

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

   10. Simplified 39.2%

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

   11. Taylor expanded in y around 0 38.1%

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

       1. associate-/l* 40.5%

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

       2. +-commutative 40.5%

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

   13. Simplified 40.5%

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

   if 1.05000000000000001e-199 < y < 1.1e-146

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 43.1%

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

      1. div-inv 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-/r* 43.1%

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

      4. metadata-eval 43.1%

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

      5. unpow2 43.1%

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

      6. associate-*l* 43.1%

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

   5. Applied egg-rr 43.1%

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

   if 1.1e-146 < y < 3.1999999999999999e-47

   1. Initial program 94.4%

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

      1. add-sqr-sqrt 94.4%

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

      2. difference-of-squares 99.7%

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

      3. hypot-def 99.7%

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

      4. hypot-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 84.4%

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

      1. associate-/l* 84.4%

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

      2. +-commutative 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in x around 0 49.0%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

   10. Simplified 49.0%

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

   11. Taylor expanded in y around 0 49.0%

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

       1. +-commutative 49.0%

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

       2. associate-*r/ 54.1%

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

   13. Simplified 54.1%

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

   if 3.1999999999999999e-47 < y < 4.00000000000000017e93

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. unpow2 35.6%

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

      2. times-frac 44.4%

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

   5. Applied egg-rr 44.4%

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

   if 4.00000000000000017e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-199}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z + x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-199}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot \left(z + x\right)}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-46}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.7e-199)
   (* -0.5 (/ (* z (+ z x)) y))
   (if (<= y 1.45e-146)
     (* x (* x (/ 0.5 y)))
     (if (<= y 6e-46)
       (* -0.5 (* z (/ (+ z x) y)))
       (if (<= y 2.5e+93) (* (/ x y) (/ x 2.0)) (* 0.5 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.7e-199) {
		tmp = -0.5 * ((z * (z + x)) / y);
	} else if (y <= 1.45e-146) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 6e-46) {
		tmp = -0.5 * (z * ((z + x) / y));
	} else if (y <= 2.5e+93) {
		tmp = (x / y) * (x / 2.0);
	} 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 <= 3.7d-199) then
        tmp = (-0.5d0) * ((z * (z + x)) / y)
    else if (y <= 1.45d-146) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 6d-46) then
        tmp = (-0.5d0) * (z * ((z + x) / y))
    else if (y <= 2.5d+93) then
        tmp = (x / y) * (x / 2.0d0)
    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 <= 3.7e-199) {
		tmp = -0.5 * ((z * (z + x)) / y);
	} else if (y <= 1.45e-146) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 6e-46) {
		tmp = -0.5 * (z * ((z + x) / y));
	} else if (y <= 2.5e+93) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.7e-199:
		tmp = -0.5 * ((z * (z + x)) / y)
	elif y <= 1.45e-146:
		tmp = x * (x * (0.5 / y))
	elif y <= 6e-46:
		tmp = -0.5 * (z * ((z + x) / y))
	elif y <= 2.5e+93:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.7e-199)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z + x)) / y));
	elseif (y <= 1.45e-146)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 6e-46)
		tmp = Float64(-0.5 * Float64(z * Float64(Float64(z + x) / y)));
	elseif (y <= 2.5e+93)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.7e-199)
		tmp = -0.5 * ((z * (z + x)) / y);
	elseif (y <= 1.45e-146)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 6e-46)
		tmp = -0.5 * (z * ((z + x) / y));
	elseif (y <= 2.5e+93)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.7e-199], N[(-0.5 * N[(N[(z * N[(z + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e-146], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-46], N[(-0.5 * N[(z * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+93], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 3.69999999999999999e-199

   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. add-sqr-sqrt 74.6%

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

      2. difference-of-squares 77.0%

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

      3. hypot-def 78.3%

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

      4. hypot-def 79.7%

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

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in y around 0 64.6%

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

      1. associate-/l* 67.8%

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

      2. +-commutative 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around 0 39.2%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 39.2%

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

      2. neg-mul-1 39.2%

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

   10. Simplified 39.2%

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

   11. Taylor expanded in y around 0 38.1%

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

   if 3.69999999999999999e-199 < y < 1.45000000000000005e-146

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 43.1%

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

      1. div-inv 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-/r* 43.1%

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

      4. metadata-eval 43.1%

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

      5. unpow2 43.1%

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

      6. associate-*l* 43.1%

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

   5. Applied egg-rr 43.1%

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

   if 1.45000000000000005e-146 < y < 5.99999999999999975e-46

   1. Initial program 94.4%

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

      1. add-sqr-sqrt 94.4%

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

      2. difference-of-squares 99.7%

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

      3. hypot-def 99.7%

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

      4. hypot-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 84.4%

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

      1. associate-/l* 84.4%

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

      2. +-commutative 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in x around 0 49.0%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

   10. Simplified 49.0%

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

   11. Taylor expanded in y around 0 49.0%

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

       1. +-commutative 49.0%

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

       2. associate-*r/ 54.1%

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

   13. Simplified 54.1%

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

   if 5.99999999999999975e-46 < y < 2.5000000000000001e93

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. unpow2 35.6%

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

      2. times-frac 44.4%

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

   5. Applied egg-rr 44.4%

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

   if 2.5000000000000001e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-199}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot \left(z + x\right)}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-146}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-46}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z + x}{y}\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-241}:\\ \;\;\;\;0.5 \cdot \frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e-264)
   (/ x (* 2.0 (/ y x)))
   (if (<= y 1.75e-241)
     (* 0.5 (/ (- x) (/ y z)))
     (if (<= y 2.3e+93) (* (/ x y) (/ x 2.0)) (* 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-264) {
		tmp = x / (2.0 * (y / x));
	} else if (y <= 1.75e-241) {
		tmp = 0.5 * (-x / (y / z));
	} else if (y <= 2.3e+93) {
		tmp = (x / y) * (x / 2.0);
	} 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 <= 3.1d-264) then
        tmp = x / (2.0d0 * (y / x))
    else if (y <= 1.75d-241) then
        tmp = 0.5d0 * (-x / (y / z))
    else if (y <= 2.3d+93) then
        tmp = (x / y) * (x / 2.0d0)
    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 <= 3.1e-264) {
		tmp = x / (2.0 * (y / x));
	} else if (y <= 1.75e-241) {
		tmp = 0.5 * (-x / (y / z));
	} else if (y <= 2.3e+93) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.1e-264:
		tmp = x / (2.0 * (y / x))
	elif y <= 1.75e-241:
		tmp = 0.5 * (-x / (y / z))
	elif y <= 2.3e+93:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e-264)
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	elseif (y <= 1.75e-241)
		tmp = Float64(0.5 * Float64(Float64(-x) / Float64(y / z)));
	elseif (y <= 2.3e+93)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.1e-264)
		tmp = x / (2.0 * (y / x));
	elseif (y <= 1.75e-241)
		tmp = 0.5 * (-x / (y / z));
	elseif (y <= 2.3e+93)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.1e-264], N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-241], N[(0.5 * N[((-x) / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+93], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.1000000000000002e-264

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf 32.2%

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

      1. unpow2 32.2%

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

      2. times-frac 34.4%

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

   5. Applied egg-rr 34.4%

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

      1. clear-num 34.4%

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

      2. frac-times 34.4%

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

      3. *-un-lft-identity 34.4%

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

   7. Applied egg-rr 34.4%

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

   if 3.1000000000000002e-264 < y < 1.7499999999999999e-241

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. difference-of-squares 100.0%

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

      3. hypot-def 100.0%

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

      4. hypot-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto 0.5 \cdot \frac{z + x}{\color{blue}{-1 \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in z around 0 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 27.2%

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

       2. associate-/l* 51.0%

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

   13. Simplified 51.0%

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

   if 1.7499999999999999e-241 < y < 2.3000000000000002e93

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 40.1%

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

      1. unpow2 40.1%

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

      2. times-frac 42.9%

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

   5. Applied egg-rr 42.9%

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

   if 2.3000000000000002e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-241}:\\ \;\;\;\;0.5 \cdot \frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y + z}{\frac{-y}{z}}}{2}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-48)
   (/ (/ (+ y z) (/ (- y) z)) 2.0)
   (if (<= y 3.3e+93) (* (/ x y) (/ x 2.0)) (* 0.5 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-48) {
		tmp = ((y + z) / (-y / z)) / 2.0;
	} else if (y <= 3.3e+93) {
		tmp = (x / y) * (x / 2.0);
	} 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 <= 9.2d-48) then
        tmp = ((y + z) / (-y / z)) / 2.0d0
    else if (y <= 3.3d+93) then
        tmp = (x / y) * (x / 2.0d0)
    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 <= 9.2e-48) {
		tmp = ((y + z) / (-y / z)) / 2.0;
	} else if (y <= 3.3e+93) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e-48:
		tmp = ((y + z) / (-y / z)) / 2.0
	elif y <= 3.3e+93:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e-48)
		tmp = Float64(Float64(Float64(y + z) / Float64(Float64(-y) / z)) / 2.0);
	elseif (y <= 3.3e+93)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e-48)
		tmp = ((y + z) / (-y / z)) / 2.0;
	elseif (y <= 3.3e+93)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e-48], N[(N[(N[(y + z), $MachinePrecision] / N[((-y) / z), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[y, 3.3e+93], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2000000000000003e-48

   1. Initial program 78.1%

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

      1. div-sub 74.4%

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

      2. sub-neg 74.4%

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

      3. div-inv 74.3%

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

      4. add-sqr-sqrt 74.3%

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

      5. pow2 74.3%

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

      6. hypot-def 74.3%

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

      7. *-commutative 74.3%

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

      8. associate-/r* 74.3%

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

      9. metadata-eval 74.3%

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

      10. div-inv 74.3%

          \[\leadsto {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y} + \left(-\color{blue}{\left(z \cdot z\right) \cdot \frac{1}{y \cdot 2}}\right) \]

      11. pow2 74.3%

          \[\leadsto {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y} + \left(-\color{blue}{{z}^{2}} \cdot \frac{1}{y \cdot 2}\right) \]

      12. *-commutative 74.3%

          \[\leadsto {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y} + \left(-{z}^{2} \cdot \frac{1}{\color{blue}{2 \cdot y}}\right) \]

      13. associate-/r* 74.3%

          \[\leadsto {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \frac{0.5}{y} + \left(-{z}^{2} \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right) \]

      14. metadata-eval 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. sub-neg 74.3%

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

      2. distribute-rgt-out-- 78.1%

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

   6. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r/ 78.1%

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

      3. associate-/l* 78.1%

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

      4. div-inv 78.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}{\color{blue}{y \cdot \frac{1}{0.5}}} \]

      5. metadata-eval 78.1%

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

      6. unpow2 78.1%

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

      7. unpow2 78.1%

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

      8. difference-of-squares 83.3%

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

      9. times-frac 99.9%

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

      10. associate-*r/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 50.8%

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

       1. associate-/l* 65.8%

          \[\leadsto \frac{\color{blue}{\frac{y + z}{\frac{y}{y - z}}}}{2} \]

   11. Simplified 65.8%

       \[\leadsto \frac{\color{blue}{\frac{y + z}{\frac{y}{y - z}}}}{2} \]

   12. Taylor expanded in y around 0 37.9%

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

       1. associate-*r/ 37.9%

          \[\leadsto \frac{\frac{y + z}{\color{blue}{\frac{-1 \cdot y}{z}}}}{2} \]

       2. mul-1-neg 37.9%

          \[\leadsto \frac{\frac{y + z}{\frac{\color{blue}{-y}}{z}}}{2} \]

   14. Simplified 37.9%

       \[\leadsto \frac{\frac{y + z}{\color{blue}{\frac{-y}{z}}}}{2} \]

   if 9.2000000000000003e-48 < y < 3.30000000000000009e93

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. unpow2 35.6%

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

      2. times-frac 44.4%

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

   5. Applied egg-rr 44.4%

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

   if 3.30000000000000009e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y + z}{\frac{-y}{z}}}{2}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \frac{\left(z + x\right) \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e+94) (* 0.5 (/ (* (+ z x) (- x z)) y)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e+94) {
		tmp = 0.5 * (((z + x) * (x - z)) / 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 <= 3.1d+94) then
        tmp = 0.5d0 * (((z + x) * (x - z)) / 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 <= 3.1e+94) {
		tmp = 0.5 * (((z + x) * (x - z)) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.1e+94:
		tmp = 0.5 * (((z + x) * (x - z)) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e+94)
		tmp = Float64(0.5 * Float64(Float64(Float64(z + x) * Float64(x - z)) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.1e+94)
		tmp = 0.5 * (((z + x) * (x - z)) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.1e+94], N[(0.5 * N[(N[(N[(z + x), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999991e94

   1. Initial program 79.4%

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

      1. add-sqr-sqrt 79.4%

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

      2. difference-of-squares 82.1%

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

      3. hypot-def 83.1%

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

      4. hypot-def 84.1%

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

   4. Applied egg-rr 84.1%

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

   5. Taylor expanded in y around 0 67.6%

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

   if 3.09999999999999991e94 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \frac{\left(z + x\right) \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \frac{\left(z + x\right) \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + z}{\frac{y}{y - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.25e+51)
   (* 0.5 (/ (* (+ z x) (- x z)) y))
   (* 0.5 (/ (+ y z) (/ y (- y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e+51) {
		tmp = 0.5 * (((z + x) * (x - z)) / y);
	} else {
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	}
	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 <= 2.25d+51) then
        tmp = 0.5d0 * (((z + x) * (x - z)) / y)
    else
        tmp = 0.5d0 * ((y + z) / (y / (y - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e+51) {
		tmp = 0.5 * (((z + x) * (x - z)) / y);
	} else {
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.25e+51:
		tmp = 0.5 * (((z + x) * (x - z)) / y)
	else:
		tmp = 0.5 * ((y + z) / (y / (y - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.25e+51)
		tmp = Float64(0.5 * Float64(Float64(Float64(z + x) * Float64(x - z)) / y));
	else
		tmp = Float64(0.5 * Float64(Float64(y + z) / Float64(y / Float64(y - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.25e+51)
		tmp = 0.5 * (((z + x) * (x - z)) / y);
	else
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.25e+51], N[(0.5 * N[(N[(N[(z + x), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + z), $MachinePrecision] / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e51

   1. Initial program 79.4%

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

      1. add-sqr-sqrt 79.4%

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

      2. difference-of-squares 82.2%

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

      3. hypot-def 83.2%

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

      4. hypot-def 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in y around 0 68.9%

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

   if 2.25e51 < y

   1. Initial program 43.7%

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

      1. add-sqr-sqrt 43.7%

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

      2. difference-of-squares 44.4%

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

      3. hypot-def 46.2%

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

      4. hypot-def 48.1%

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

   4. Applied egg-rr 48.1%

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

   5. Taylor expanded in x around 0 39.8%

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

      1. associate-/l* 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \frac{\left(z + x\right) \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + z}{\frac{y}{y - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \frac{z + x}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + z}{\frac{y}{y - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.85e+97)
   (* 0.5 (/ (+ z x) (/ y (- x z))))
   (* 0.5 (/ (+ y z) (/ y (- y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.85e+97) {
		tmp = 0.5 * ((z + x) / (y / (x - z)));
	} else {
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	}
	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 <= 2.85d+97) then
        tmp = 0.5d0 * ((z + x) / (y / (x - z)))
    else
        tmp = 0.5d0 * ((y + z) / (y / (y - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.85e+97) {
		tmp = 0.5 * ((z + x) / (y / (x - z)));
	} else {
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.85e+97:
		tmp = 0.5 * ((z + x) / (y / (x - z)))
	else:
		tmp = 0.5 * ((y + z) / (y / (y - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.85e+97)
		tmp = Float64(0.5 * Float64(Float64(z + x) / Float64(y / Float64(x - z))));
	else
		tmp = Float64(0.5 * Float64(Float64(y + z) / Float64(y / Float64(y - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.85e+97)
		tmp = 0.5 * ((z + x) / (y / (x - z)));
	else
		tmp = 0.5 * ((y + z) / (y / (y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.85e+97], N[(0.5 * N[(N[(z + x), $MachinePrecision] / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + z), $MachinePrecision] / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8500000000000001e97

   1. Initial program 79.6%

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

      1. add-sqr-sqrt 79.6%

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

      2. difference-of-squares 82.3%

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

      3. hypot-def 83.3%

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

      4. hypot-def 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. associate-/l* 70.6%

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

      2. +-commutative 70.6%

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

   7. Simplified 70.6%

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

   if 2.8500000000000001e97 < y

   1. Initial program 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. difference-of-squares 34.2%

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

      3. hypot-def 36.4%

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

      4. hypot-def 38.8%

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

   4. Applied egg-rr 38.8%

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

   5. Taylor expanded in x around 0 34.7%

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

      1. associate-/l* 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \frac{z + x}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y + z}{\frac{y}{y - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+93}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6e+93) (* x (* x (/ 0.5 y))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+93) {
		tmp = x * (x * (0.5 / 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 <= 2.6d+93) then
        tmp = x * (x * (0.5d0 / 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 <= 2.6e+93) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.6e+93:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.6e+93)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.6e+93)
		tmp = x * (x * (0.5 / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.6e+93], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e93

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf 34.2%

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

      1. div-inv 34.2%

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

      2. *-commutative 34.2%

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

      3. associate-/r* 34.2%

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

      4. metadata-eval 34.2%

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

      5. unpow2 34.2%

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

      6. associate-*l* 36.5%

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

   5. Applied egg-rr 36.5%

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

   if 2.6e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+93}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e+93) (* (/ x y) (/ x 2.0)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e+93) {
		tmp = (x / y) * (x / 2.0);
	} 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 <= 3.1d+93) then
        tmp = (x / y) * (x / 2.0d0)
    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 <= 3.1e+93) {
		tmp = (x / y) * (x / 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.1e+93:
		tmp = (x / y) * (x / 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e+93)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.1e+93)
		tmp = (x / y) * (x / 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.1e+93], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000019e93

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf 34.2%

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

      1. unpow2 34.2%

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

      2. times-frac 36.5%

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

   5. Applied egg-rr 36.5%

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

   if 3.10000000000000019e93 < y

   1. Initial program 36.1%

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

   3. Taylor expanded in y around inf 78.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.6% 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 71.1%

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

3. Taylor expanded in y around inf 40.0%

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

4. Final simplification 40.0%

   \[\leadsto 0.5 \cdot y \]
5. 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 2024020 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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