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

Percentage Accurate: 69.5% → 99.9%

Time: 13.0s

Alternatives: 8

Speedup: 1.2×

• 41.5% 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: 69.5% 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

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

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (* 0.5 (+ y (* (/ (+ z_m x_m) y) (- x_m z_m)))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * (y + (((z_m + x_m) / y) * (x_m - z_m)));
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * (y + (((z_m + x_m) / y) * (x_m - z_m)))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * N[(y + N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. remove-double-neg 65.7%

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

   2. distribute-lft-neg-out 65.7%

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

   3. distribute-frac-neg2 65.7%

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

   4. distribute-frac-neg 65.7%

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

   5. neg-mul-1 65.7%

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

   6. distribute-lft-neg-out 65.7%

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

   7. *-commutative 65.7%

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

   8. distribute-lft-neg-in 65.7%

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

   9. times-frac 65.7%

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

   10. metadata-eval 65.7%

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

   11. metadata-eval 65.7%

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

   12. associate--l+ 65.7%

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

   13. fma-define 68.0%

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

3. Simplified 68.0%

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

   1. fma-undefine 65.7%

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

   2. associate--l+ 65.7%

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

   3. div-sub 61.8%

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

   4. add-sqr-sqrt 61.8%

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

   5. pow2 61.8%

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

   6. hypot-define 61.8%

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

   7. pow2 61.8%

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

6. Applied egg-rr 61.8%

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

   1. div-sub 65.7%

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

   2. unpow2 65.7%

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

   3. hypot-undefine 65.7%

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

   4. unpow2 65.7%

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

   5. unpow2 65.7%

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

   6. hypot-undefine 65.7%

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

   7. unpow2 65.7%

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

   8. unpow2 65.7%

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

   9. rem-square-sqrt 65.7%

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

   10. associate-+r- 65.7%

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

   11. +-commutative 65.7%

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

   12. associate-+l- 65.7%

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

   13. div-sub 65.7%

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

   14. unpow2 65.7%

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

   15. associate-/l* 84.5%

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

   16. *-inverses 84.5%

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

   17. *-rgt-identity 84.5%

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

8. Simplified 84.5%

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

   1. unpow2 84.5%

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

   2. unpow2 84.5%

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

   3. difference-of-squares 90.4%

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

10. Applied egg-rr 90.4%

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

    1. *-commutative 90.4%

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

    2. associate-/l* 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 52.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{z\_m}{y} \cdot \frac{z\_m}{-2}\\ \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 5 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x\_m \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 1.04 \cdot 10^{+80}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* (/ z_m y) (/ z_m -2.0))))
   (if (<= x_m 5.2e-280)
     t_0
     (if (<= x_m 5e-125)
       (* 0.5 y)
       (if (<= x_m 1.2e-65)
         t_0
         (if (<= x_m 1.04e+80) (* 0.5 y) (* x_m (* x_m (/ 0.5 y)))))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double t_0 = (z_m / y) * (z_m / -2.0);
	double tmp;
	if (x_m <= 5.2e-280) {
		tmp = t_0;
	} else if (x_m <= 5e-125) {
		tmp = 0.5 * y;
	} else if (x_m <= 1.2e-65) {
		tmp = t_0;
	} else if (x_m <= 1.04e+80) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * (x_m * (0.5 / y));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z_m / y) * (z_m / (-2.0d0))
    if (x_m <= 5.2d-280) then
        tmp = t_0
    else if (x_m <= 5d-125) then
        tmp = 0.5d0 * y
    else if (x_m <= 1.2d-65) then
        tmp = t_0
    else if (x_m <= 1.04d+80) then
        tmp = 0.5d0 * y
    else
        tmp = x_m * (x_m * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double t_0 = (z_m / y) * (z_m / -2.0);
	double tmp;
	if (x_m <= 5.2e-280) {
		tmp = t_0;
	} else if (x_m <= 5e-125) {
		tmp = 0.5 * y;
	} else if (x_m <= 1.2e-65) {
		tmp = t_0;
	} else if (x_m <= 1.04e+80) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * (x_m * (0.5 / y));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	t_0 = (z_m / y) * (z_m / -2.0)
	tmp = 0
	if x_m <= 5.2e-280:
		tmp = t_0
	elif x_m <= 5e-125:
		tmp = 0.5 * y
	elif x_m <= 1.2e-65:
		tmp = t_0
	elif x_m <= 1.04e+80:
		tmp = 0.5 * y
	else:
		tmp = x_m * (x_m * (0.5 / y))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(z_m / y) * Float64(z_m / -2.0))
	tmp = 0.0
	if (x_m <= 5.2e-280)
		tmp = t_0;
	elseif (x_m <= 5e-125)
		tmp = Float64(0.5 * y);
	elseif (x_m <= 1.2e-65)
		tmp = t_0;
	elseif (x_m <= 1.04e+80)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x_m * Float64(x_m * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (z_m / y) * (z_m / -2.0);
	tmp = 0.0;
	if (x_m <= 5.2e-280)
		tmp = t_0;
	elseif (x_m <= 5e-125)
		tmp = 0.5 * y;
	elseif (x_m <= 1.2e-65)
		tmp = t_0;
	elseif (x_m <= 1.04e+80)
		tmp = 0.5 * y;
	else
		tmp = x_m * (x_m * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m / y), $MachinePrecision] * N[(z$95$m / -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.2e-280], t$95$0, If[LessEqual[x$95$m, 5e-125], N[(0.5 * y), $MachinePrecision], If[LessEqual[x$95$m, 1.2e-65], t$95$0, If[LessEqual[x$95$m, 1.04e+80], N[(0.5 * y), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.2e-280 or 4.99999999999999967e-125 < x < 1.2000000000000001e-65

   1. Initial program 66.8%

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

      1. clear-num 66.8%

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

      2. inv-pow 66.8%

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

      3. associate-/l* 65.9%

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

      4. add-sqr-sqrt 65.9%

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

      5. pow2 65.9%

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

      6. hypot-define 65.9%

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

      7. pow2 65.9%

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

   4. Applied egg-rr 65.9%

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

   5. Taylor expanded in z around inf 30.8%

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

      1. associate-*r/ 30.8%

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

      2. *-commutative 30.8%

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

   7. Simplified 30.8%

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

      1. unpow-1 30.8%

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

      2. clear-num 30.8%

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

      3. unpow2 30.8%

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

      4. times-frac 34.6%

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

   9. Applied egg-rr 34.6%

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

   if 5.2e-280 < x < 4.99999999999999967e-125 or 1.2000000000000001e-65 < x < 1.04000000000000006e80

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf 52.7%

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

      1. *-commutative 52.7%

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

   5. Simplified 52.7%

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

   if 1.04000000000000006e80 < x

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*l/ 65.3%

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

   5. Simplified 65.3%

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

      1. associate-/l* 65.3%

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

      2. unpow2 65.3%

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

      3. associate-*l* 70.5%

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

   7. Applied egg-rr 70.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{z}{-2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{z}{-2}\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{+80}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{z\_m}{y} \cdot \frac{z\_m}{-2}\\ \mathbf{if}\;x\_m \leq 1.3 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 4 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x\_m \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 10^{+80}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{0.5 \cdot x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* (/ z_m y) (/ z_m -2.0))))
   (if (<= x_m 1.3e-280)
     t_0
     (if (<= x_m 4e-125)
       (* 0.5 y)
       (if (<= x_m 1.2e-65)
         t_0
         (if (<= x_m 1e+80) (* 0.5 y) (* x_m (/ (* 0.5 x_m) y))))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double t_0 = (z_m / y) * (z_m / -2.0);
	double tmp;
	if (x_m <= 1.3e-280) {
		tmp = t_0;
	} else if (x_m <= 4e-125) {
		tmp = 0.5 * y;
	} else if (x_m <= 1.2e-65) {
		tmp = t_0;
	} else if (x_m <= 1e+80) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * ((0.5 * x_m) / y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z_m / y) * (z_m / (-2.0d0))
    if (x_m <= 1.3d-280) then
        tmp = t_0
    else if (x_m <= 4d-125) then
        tmp = 0.5d0 * y
    else if (x_m <= 1.2d-65) then
        tmp = t_0
    else if (x_m <= 1d+80) then
        tmp = 0.5d0 * y
    else
        tmp = x_m * ((0.5d0 * x_m) / y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double t_0 = (z_m / y) * (z_m / -2.0);
	double tmp;
	if (x_m <= 1.3e-280) {
		tmp = t_0;
	} else if (x_m <= 4e-125) {
		tmp = 0.5 * y;
	} else if (x_m <= 1.2e-65) {
		tmp = t_0;
	} else if (x_m <= 1e+80) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * ((0.5 * x_m) / y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	t_0 = (z_m / y) * (z_m / -2.0)
	tmp = 0
	if x_m <= 1.3e-280:
		tmp = t_0
	elif x_m <= 4e-125:
		tmp = 0.5 * y
	elif x_m <= 1.2e-65:
		tmp = t_0
	elif x_m <= 1e+80:
		tmp = 0.5 * y
	else:
		tmp = x_m * ((0.5 * x_m) / y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(z_m / y) * Float64(z_m / -2.0))
	tmp = 0.0
	if (x_m <= 1.3e-280)
		tmp = t_0;
	elseif (x_m <= 4e-125)
		tmp = Float64(0.5 * y);
	elseif (x_m <= 1.2e-65)
		tmp = t_0;
	elseif (x_m <= 1e+80)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x_m * Float64(Float64(0.5 * x_m) / y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (z_m / y) * (z_m / -2.0);
	tmp = 0.0;
	if (x_m <= 1.3e-280)
		tmp = t_0;
	elseif (x_m <= 4e-125)
		tmp = 0.5 * y;
	elseif (x_m <= 1.2e-65)
		tmp = t_0;
	elseif (x_m <= 1e+80)
		tmp = 0.5 * y;
	else
		tmp = x_m * ((0.5 * x_m) / y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m / y), $MachinePrecision] * N[(z$95$m / -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.3e-280], t$95$0, If[LessEqual[x$95$m, 4e-125], N[(0.5 * y), $MachinePrecision], If[LessEqual[x$95$m, 1.2e-65], t$95$0, If[LessEqual[x$95$m, 1e+80], N[(0.5 * y), $MachinePrecision], N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3e-280 or 4.00000000000000005e-125 < x < 1.2000000000000001e-65

   1. Initial program 66.8%

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

      1. clear-num 66.8%

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

      2. inv-pow 66.8%

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

      3. associate-/l* 65.9%

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

      4. add-sqr-sqrt 65.9%

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

      5. pow2 65.9%

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

      6. hypot-define 65.9%

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

      7. pow2 65.9%

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

   4. Applied egg-rr 65.9%

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

   5. Taylor expanded in z around inf 30.8%

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

      1. associate-*r/ 30.8%

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

      2. *-commutative 30.8%

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

   7. Simplified 30.8%

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

      1. unpow-1 30.8%

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

      2. clear-num 30.8%

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

      3. unpow2 30.8%

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

      4. times-frac 34.6%

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

   9. Applied egg-rr 34.6%

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

   if 1.3e-280 < x < 4.00000000000000005e-125 or 1.2000000000000001e-65 < x < 1e80

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf 52.7%

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

      1. *-commutative 52.7%

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

   5. Simplified 52.7%

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

   if 1e80 < x

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*l/ 65.3%

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

   5. Simplified 65.3%

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

      1. clear-num 65.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

      2. inv-pow 65.2%

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

      3. *-un-lft-identity 65.2%

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

      4. *-commutative 65.2%

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

      5. times-frac 65.2%

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

      6. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. unpow-1 65.2%

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

   9. Simplified 65.2%

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

       1. associate-*r/ 65.2%

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

       2. unpow2 65.2%

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

       3. associate-/r* 70.4%

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

       4. *-commutative 70.4%

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

   11. Applied egg-rr 70.4%

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

       1. associate-/l/ 65.2%

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

       2. associate-/r/ 65.3%

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

       3. metadata-eval 65.3%

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

       4. div-inv 65.3%

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

       5. clear-num 65.3%

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

       6. associate-*l* 70.5%

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

       7. *-commutative 70.5%

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

       8. associate-*r/ 70.5%

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

   13. Applied egg-rr 70.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{z}{-2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{z}{-2}\\ \mathbf{elif}\;x \leq 10^{+80}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{+103} \lor \neg \left(z\_m \leq 5.6 \cdot 10^{+118}\right) \land z\_m \leq 1.05 \cdot 10^{+182}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x\_m}{y} \cdot \left(x\_m - z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{y} \cdot \frac{z\_m}{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (or (<= z_m 2.5e+103) (and (not (<= z_m 5.6e+118)) (<= z_m 1.05e+182)))
   (* 0.5 (+ y (* (/ x_m y) (- x_m z_m))))
   (* (/ z_m y) (/ z_m -2.0))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((z_m <= 2.5e+103) || (!(z_m <= 5.6e+118) && (z_m <= 1.05e+182))) {
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	} else {
		tmp = (z_m / y) * (z_m / -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m <= 2.5d+103) .or. (.not. (z_m <= 5.6d+118)) .and. (z_m <= 1.05d+182)) then
        tmp = 0.5d0 * (y + ((x_m / y) * (x_m - z_m)))
    else
        tmp = (z_m / y) * (z_m / (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((z_m <= 2.5e+103) || (!(z_m <= 5.6e+118) && (z_m <= 1.05e+182))) {
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	} else {
		tmp = (z_m / y) * (z_m / -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (z_m <= 2.5e+103) or (not (z_m <= 5.6e+118) and (z_m <= 1.05e+182)):
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)))
	else:
		tmp = (z_m / y) * (z_m / -2.0)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if ((z_m <= 2.5e+103) || (!(z_m <= 5.6e+118) && (z_m <= 1.05e+182)))
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x_m / y) * Float64(x_m - z_m))));
	else
		tmp = Float64(Float64(z_m / y) * Float64(z_m / -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((z_m <= 2.5e+103) || (~((z_m <= 5.6e+118)) && (z_m <= 1.05e+182)))
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	else
		tmp = (z_m / y) * (z_m / -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[Or[LessEqual[z$95$m, 2.5e+103], And[N[Not[LessEqual[z$95$m, 5.6e+118]], $MachinePrecision], LessEqual[z$95$m, 1.05e+182]]], N[(0.5 * N[(y + N[(N[(x$95$m / y), $MachinePrecision] * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / y), $MachinePrecision] * N[(z$95$m / -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e103 or 5.59999999999999972e118 < z < 1.0499999999999999e182

   1. Initial program 66.7%

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

      1. remove-double-neg 66.7%

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

      2. distribute-lft-neg-out 66.7%

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

      3. distribute-frac-neg2 66.7%

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

      4. distribute-frac-neg 66.7%

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

      5. neg-mul-1 66.7%

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

      6. distribute-lft-neg-out 66.7%

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

      7. *-commutative 66.7%

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

      8. distribute-lft-neg-in 66.7%

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

      9. times-frac 66.7%

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

      10. metadata-eval 66.7%

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

      11. metadata-eval 66.7%

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

      12. associate--l+ 66.7%

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

      13. fma-define 68.0%

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

   3. Simplified 68.0%

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

      1. fma-undefine 66.7%

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

      2. associate--l+ 66.7%

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

      3. div-sub 62.7%

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

      4. add-sqr-sqrt 62.7%

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

      5. pow2 62.7%

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

      6. hypot-define 62.7%

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

      7. pow2 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. div-sub 66.7%

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

      2. unpow2 66.7%

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

      3. hypot-undefine 66.7%

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

      4. unpow2 66.7%

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

      5. unpow2 66.7%

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

      6. hypot-undefine 66.7%

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

      7. unpow2 66.7%

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

      8. unpow2 66.7%

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

      9. rem-square-sqrt 66.7%

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

      10. associate-+r- 66.7%

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

      11. +-commutative 66.7%

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

      12. associate-+l- 66.7%

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

      13. div-sub 66.7%

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

      14. unpow2 66.7%

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

      15. associate-/l* 87.1%

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

      16. *-inverses 87.1%

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

      17. *-rgt-identity 87.1%

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

   8. Simplified 87.1%

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

      1. unpow2 87.1%

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

      2. unpow2 87.1%

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

      3. difference-of-squares 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. *-commutative 91.1%

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

       2. associate-/l* 99.9%

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

   12. Applied egg-rr 99.9%

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

   13. Taylor expanded in z around 0 79.3%

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

   if 2.5e103 < z < 5.59999999999999972e118 or 1.0499999999999999e182 < z

   1. Initial program 58.5%

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

      1. clear-num 58.5%

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

      2. inv-pow 58.5%

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

      3. associate-/l* 58.5%

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

      4. add-sqr-sqrt 58.5%

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

      5. pow2 58.5%

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

      6. hypot-define 58.5%

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

      7. pow2 58.5%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in z around inf 75.4%

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

      1. associate-*r/ 75.4%

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

      2. *-commutative 75.4%

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

   7. Simplified 75.4%

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

      1. unpow-1 75.4%

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

      2. clear-num 75.4%

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

      3. unpow2 75.4%

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

      4. times-frac 87.1%

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

   9. Applied egg-rr 87.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+103} \lor \neg \left(z \leq 5.6 \cdot 10^{+118}\right) \land z \leq 1.05 \cdot 10^{+182}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{y} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{z}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x\_m}{y} \cdot \left(x\_m - z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{z\_m}{y} \cdot \left(x\_m - z\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= z_m 4.1e+21)
   (* 0.5 (+ y (* (/ x_m y) (- x_m z_m))))
   (* 0.5 (+ y (* (/ z_m y) (- x_m z_m))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 4.1e+21) {
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	} else {
		tmp = 0.5 * (y + ((z_m / y) * (x_m - z_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 4.1d+21) then
        tmp = 0.5d0 * (y + ((x_m / y) * (x_m - z_m)))
    else
        tmp = 0.5d0 * (y + ((z_m / y) * (x_m - z_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (z_m <= 4.1e+21) {
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	} else {
		tmp = 0.5 * (y + ((z_m / y) * (x_m - z_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if z_m <= 4.1e+21:
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)))
	else:
		tmp = 0.5 * (y + ((z_m / y) * (x_m - z_m)))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (z_m <= 4.1e+21)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x_m / y) * Float64(x_m - z_m))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(z_m / y) * Float64(x_m - z_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (z_m <= 4.1e+21)
		tmp = 0.5 * (y + ((x_m / y) * (x_m - z_m)));
	else
		tmp = 0.5 * (y + ((z_m / y) * (x_m - z_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[z$95$m, 4.1e+21], N[(0.5 * N[(y + N[(N[(x$95$m / y), $MachinePrecision] * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(z$95$m / y), $MachinePrecision] * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.1e21

   1. Initial program 67.6%

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

      1. remove-double-neg 67.6%

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

      2. distribute-lft-neg-out 67.6%

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

      3. distribute-frac-neg2 67.6%

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

      4. distribute-frac-neg 67.6%

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

      5. neg-mul-1 67.6%

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

      6. distribute-lft-neg-out 67.6%

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

      7. *-commutative 67.6%

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

      8. distribute-lft-neg-in 67.6%

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

      9. times-frac 67.6%

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

      10. metadata-eval 67.6%

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

      11. metadata-eval 67.6%

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

      12. associate--l+ 67.6%

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

      13. fma-define 69.1%

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

   3. Simplified 69.1%

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

      1. fma-undefine 67.6%

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

      2. associate--l+ 67.6%

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

      3. div-sub 65.6%

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

      4. add-sqr-sqrt 65.6%

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

      5. pow2 65.6%

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

      6. hypot-define 65.6%

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

      7. pow2 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. div-sub 67.6%

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

      2. unpow2 67.6%

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

      3. hypot-undefine 67.6%

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

      4. unpow2 67.6%

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

      5. unpow2 67.6%

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

      6. hypot-undefine 67.6%

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

      7. unpow2 67.6%

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

      8. unpow2 67.6%

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

      9. rem-square-sqrt 67.6%

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

      10. associate-+r- 67.6%

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

      11. +-commutative 67.6%

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

      12. associate-+l- 67.6%

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

      13. div-sub 67.6%

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

      14. unpow2 67.6%

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

      15. associate-/l* 87.4%

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

      16. *-inverses 87.4%

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

      17. *-rgt-identity 87.4%

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

   8. Simplified 87.4%

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

      1. unpow2 87.4%

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

      2. unpow2 87.4%

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

      3. difference-of-squares 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. *-commutative 90.9%

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

       2. associate-/l* 99.9%

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

   12. Applied egg-rr 99.9%

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

   13. Taylor expanded in z around 0 78.5%

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

   if 4.1e21 < z

   1. Initial program 59.0%

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

      1. remove-double-neg 59.0%

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

      2. distribute-lft-neg-out 59.0%

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

      3. distribute-frac-neg2 59.0%

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

      4. distribute-frac-neg 59.0%

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

      5. neg-mul-1 59.0%

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

      6. distribute-lft-neg-out 59.0%

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

      7. *-commutative 59.0%

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

      8. distribute-lft-neg-in 59.0%

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

      9. times-frac 59.0%

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

      10. metadata-eval 59.0%

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

      11. metadata-eval 59.0%

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

      12. associate--l+ 59.0%

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

      13. fma-define 64.3%

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

   3. Simplified 64.3%

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

      1. fma-undefine 59.0%

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

      2. associate--l+ 59.0%

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

      3. div-sub 48.5%

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

      4. add-sqr-sqrt 48.5%

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

      5. pow2 48.5%

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

      6. hypot-define 48.5%

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

      7. pow2 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. div-sub 59.0%

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

      2. unpow2 59.0%

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

      3. hypot-undefine 59.0%

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

      4. unpow2 59.0%

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

      5. unpow2 59.0%

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

      6. hypot-undefine 59.0%

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

      7. unpow2 59.0%

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

      8. unpow2 59.0%

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

      9. rem-square-sqrt 59.0%

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

      10. associate-+r- 59.0%

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

      11. +-commutative 59.0%

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

      12. associate-+l- 59.0%

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

      13. div-sub 59.0%

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

      14. unpow2 59.0%

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

      15. associate-/l* 74.3%

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

      16. *-inverses 74.3%

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

      17. *-rgt-identity 74.3%

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

   8. Simplified 74.3%

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

      1. unpow2 74.3%

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

      2. unpow2 74.3%

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

      3. difference-of-squares 88.7%

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

   10. Applied egg-rr 88.7%

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

       1. *-commutative 88.7%

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

       2. associate-/l* 100.0%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in z around inf 83.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{y} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{z}{y} \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= x_m 8.5e+79) (* 0.5 y) (* x_m (* x_m (/ 0.5 y)))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (x_m <= 8.5e+79) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * (x_m * (0.5 / y));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 8.5d+79) then
        tmp = 0.5d0 * y
    else
        tmp = x_m * (x_m * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (x_m <= 8.5e+79) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * (x_m * (0.5 / y));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if x_m <= 8.5e+79:
		tmp = 0.5 * y
	else:
		tmp = x_m * (x_m * (0.5 / y))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (x_m <= 8.5e+79)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x_m * Float64(x_m * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (x_m <= 8.5e+79)
		tmp = 0.5 * y;
	else
		tmp = x_m * (x_m * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[x$95$m, 8.5e+79], N[(0.5 * y), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999998e79

   1. Initial program 65.8%

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

   3. Taylor expanded in y around inf 40.8%

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

      1. *-commutative 40.8%

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

   5. Simplified 40.8%

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

   if 8.4999999999999998e79 < x

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*l/ 65.3%

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

   5. Simplified 65.3%

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

      1. associate-/l* 65.3%

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

      2. unpow2 65.3%

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

      3. associate-*l* 70.5%

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

   7. Applied egg-rr 70.5%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 1.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y \cdot -0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m) :precision binary64 (* y -0.5))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	return y * -0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return y * -0.5;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return y * -0.5

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return Float64(y * -0.5)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp = code(x_m, y, z_m)
	tmp = y * -0.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(y * -0.5), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. add-sqr-sqrt 34.4%

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

   2. sqrt-unprod 32.0%

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

   3. associate-/r* 32.0%

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

   4. associate--l+ 32.0%

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

   5. fma-undefine 32.0%

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

   6. associate-/r* 32.0%

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

   7. associate--l+ 32.0%

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

   8. fma-undefine 34.4%

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

   9. frac-times 34.4%

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

4. Applied egg-rr 32.0%

   \[\leadsto \color{blue}{\sqrt{\frac{{\left(\frac{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}{y}\right)}^{2}}{4}}} \]
5. Step-by-step derivation

6. Simplified 32.7%

   \[\leadsto \color{blue}{\sqrt{\frac{{\left(y - \frac{{z}^{2} - {x}^{2}}{y}\right)}^{2}}{4}}} \]

7. Taylor expanded in y around -inf 1.7%

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

   1. *-commutative 1.7%

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

9. Simplified 1.7%

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

10. Final simplification 1.7%

    \[\leadsto y \cdot -0.5 \]
11. Add Preprocessing

Alternative 8: 34.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m) :precision binary64 (* 0.5 y))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * y

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

3. Taylor expanded in y around inf 36.4%

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

   1. *-commutative 36.4%

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

5. Simplified 36.4%

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

6. Final simplification 36.4%

   \[\leadsto 0.5 \cdot y \]
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 2024044 
(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)))