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

Percentage Accurate: 69.2% → 95.1%

Time: 10.1s

Alternatives: 13

Speedup: 1.1×

• 41.7% 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: 69.2% 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: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-18} \lor \neg \left(y \leq 3 \cdot 10^{-142}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \frac{y - z}{y} \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y - z\right) \cdot \left(y + z\right)\right)}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.35e-18) (not (<= y 3e-142)))
   (* 0.5 (+ (/ x (/ y x)) (* (/ (- y z) y) (+ y z))))
   (/ (fma x x (* (- y z) (+ y z))) (* y 2.0))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.35e-18) || !(y <= 3e-142)) {
		tmp = 0.5 * ((x / (y / x)) + (((y - z) / y) * (y + z)));
	} else {
		tmp = fma(x, x, ((y - z) * (y + z))) / (y * 2.0);
	}
	return tmp;
}

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.35e-18) || !(y <= 3e-142))
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(Float64(Float64(y - z) / y) * Float64(y + z))));
	else
		tmp = Float64(fma(x, x, Float64(Float64(y - z) * Float64(y + z))) / Float64(y * 2.0));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -1.35e-18], N[Not[LessEqual[y, 3e-142]], $MachinePrecision]], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + N[(N[(y - z), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.34999999999999994e-18 or 3.0000000000000001e-142 < y

   1. Initial program 58.8%

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

   2. Taylor expanded in x around 0 58.8%

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

      1. distribute-lft-out 58.8%

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

      2. unpow2 58.8%

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

      3. unpow2 58.8%

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

      4. difference-of-squares 60.1%

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

      5. *-commutative 60.1%

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

      6. +-commutative 60.1%

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

      7. unpow2 60.1%

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

      8. associate-/l* 64.2%

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

      9. associate-/l* 99.2%

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

      10. associate-/r/ 99.2%

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

   4. Simplified 99.2%

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

   if -1.34999999999999994e-18 < y < 3.0000000000000001e-142

   1. Initial program 94.5%

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

      1. associate--l+ 94.5%

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

      2. fma-def 95.6%

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

      3. difference-of-squares 95.6%

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

   3. Simplified 95.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-18} \lor \neg \left(y \leq 3 \cdot 10^{-142}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \frac{y - z}{y} \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y - z\right) \cdot \left(y + z\right)\right)}{y \cdot 2}\\ \end{array} \]

Alternative 2: 94.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{y}, \frac{\mathsf{hypot}\left(x, y\right)}{2}, \frac{z}{y} \cdot \frac{-z}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{x}{y}, x, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+237)
   (fma (/ (hypot x y) y) (/ (hypot x y) 2.0) (* (/ z y) (/ (- z) 2.0)))
   (* 0.5 (fma (/ x y) x y))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+237) {
		tmp = fma((hypot(x, y) / y), (hypot(x, y) / 2.0), ((z / y) * (-z / 2.0)));
	} else {
		tmp = 0.5 * fma((x / y), x, y);
	}
	return tmp;
}

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+237)
		tmp = fma(Float64(hypot(x, y) / y), Float64(hypot(x, y) / 2.0), Float64(Float64(z / y) * Float64(Float64(-z) / 2.0)));
	else
		tmp = Float64(0.5 * fma(Float64(x / y), x, y));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+237], N[(N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / y), $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[(z / y), $MachinePrecision] * N[((-z) / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999988e237

   1. Initial program 76.9%

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

      1. div-sub 75.2%

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

      2. sub-neg 75.2%

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

      3. add-sqr-sqrt 75.2%

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

      4. times-frac 75.2%

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

      5. fma-def 76.9%

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

      6. hypot-def 76.9%

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

      7. hypot-def 91.2%

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

      8. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   if 1.99999999999999988e237 < (*.f64 x x)

   1. Initial program 59.5%

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

   2. Taylor expanded in y around 0 60.8%

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

      1. distribute-lft-out 60.8%

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

      2. unpow2 60.8%

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

      3. unpow2 60.8%

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

   4. Simplified 60.8%

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

   5. Taylor expanded in x around inf 68.8%

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

      1. unpow2 68.8%

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

      2. associate-/l* 89.8%

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

   7. Simplified 89.8%

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

      1. +-commutative 89.8%

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

      2. associate-/r/ 89.8%

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

      3. fma-def 89.8%

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

   9. Applied egg-rr 89.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{y}, \frac{\mathsf{hypot}\left(x, y\right)}{2}, \frac{z}{y} \cdot \frac{-z}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{x}{y}, x, y\right)\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+222} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+295}\right):\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+132)
   (* 0.5 (- y (/ z (/ y z))))
   (if (or (<= (* x x) 5e+222) (not (<= (* x x) 2e+295)))
     (* 0.5 (+ y (/ x (/ y x))))
     (/ (- (* x x) (* z z)) (* y 2.0)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+132) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else if (((x * x) <= 5e+222) || !((x * x) <= 2e+295)) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+132) then
        tmp = 0.5d0 * (y - (z / (y / z)))
    else if (((x * x) <= 5d+222) .or. (.not. ((x * x) <= 2d+295))) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = ((x * x) - (z * z)) / (y * 2.0d0)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+132) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else if (((x * x) <= 5e+222) || !((x * x) <= 2e+295)) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+132:
		tmp = 0.5 * (y - (z / (y / z)))
	elif ((x * x) <= 5e+222) or not ((x * x) <= 2e+295):
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = ((x * x) - (z * z)) / (y * 2.0)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+132)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	elseif ((Float64(x * x) <= 5e+222) || !(Float64(x * x) <= 2e+295))
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(x * x) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+132)
		tmp = 0.5 * (y - (z / (y / z)));
	elseif (((x * x) <= 5e+222) || ~(((x * x) <= 2e+295)))
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+132], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * x), $MachinePrecision], 5e+222], N[Not[LessEqual[N[(x * x), $MachinePrecision], 2e+295]], $MachinePrecision]], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999998e132

   1. Initial program 77.7%

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

      1. div-sub 75.7%

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

      2. sub-neg 75.7%

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

      3. add-sqr-sqrt 75.7%

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

      4. times-frac 75.7%

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

      5. fma-def 77.7%

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

      6. hypot-def 77.7%

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

      7. hypot-def 93.5%

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

      8. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 85.0%

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

      1. distribute-lft-out-- 85.0%

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

      2. unpow2 85.0%

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

      3. associate-/l* 91.3%

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

   6. Simplified 91.3%

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

   if 1.99999999999999998e132 < (*.f64 x x) < 5.00000000000000023e222 or 2e295 < (*.f64 x x)

   1. Initial program 56.9%

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

   2. Taylor expanded in y around 0 60.1%

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

      1. distribute-lft-out 60.1%

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

      2. unpow2 60.1%

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

      3. unpow2 60.1%

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

   4. Simplified 60.1%

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

   5. Taylor expanded in x around inf 74.8%

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

      1. unpow2 74.8%

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

      2. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if 5.00000000000000023e222 < (*.f64 x x) < 2e295

   1. Initial program 94.5%

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

   2. Taylor expanded in y around 0 94.5%

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

      1. unpow2 94.5%

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

      2. unpow2 94.5%

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

   4. Simplified 94.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+222} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+295}\right):\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 4: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-18} \lor \neg \left(y \leq 1.65 \cdot 10^{-96}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \frac{y - z}{y} \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-18) (not (<= y 1.65e-96)))
   (* 0.5 (+ (/ x (/ y x)) (* (/ (- y z) y) (+ y z))))
   (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-18) || !(y <= 1.65e-96)) {
		tmp = 0.5 * ((x / (y / x)) + (((y - z) / y) * (y + z)));
	} else {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 <= (-1d-18)) .or. (.not. (y <= 1.65d-96))) then
        tmp = 0.5d0 * ((x / (y / x)) + (((y - z) / y) * (y + z)))
    else
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-18) || !(y <= 1.65e-96)) {
		tmp = 0.5 * ((x / (y / x)) + (((y - z) / y) * (y + z)));
	} else {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (y <= -1e-18) or not (y <= 1.65e-96):
		tmp = 0.5 * ((x / (y / x)) + (((y - z) / y) * (y + z)))
	else:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-18) || !(y <= 1.65e-96))
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(Float64(Float64(y - z) / y) * Float64(y + z))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e-18) || ~((y <= 1.65e-96)))
		tmp = 0.5 * ((x / (y / x)) + (((y - z) / y) * (y + z)));
	else
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -1e-18], N[Not[LessEqual[y, 1.65e-96]], $MachinePrecision]], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0000000000000001e-18 or 1.64999999999999995e-96 < y

   1. Initial program 55.6%

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

   2. Taylor expanded in x around 0 55.6%

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

      1. distribute-lft-out 55.6%

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

      2. unpow2 55.6%

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

      3. unpow2 55.6%

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

      4. difference-of-squares 57.0%

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

      5. *-commutative 57.0%

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

      6. +-commutative 57.0%

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

      7. unpow2 57.0%

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

      8. associate-/l* 61.4%

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

      9. associate-/l* 99.2%

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

      10. associate-/r/ 99.2%

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

   4. Simplified 99.2%

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

   if -1.0000000000000001e-18 < y < 1.64999999999999995e-96

   1. Initial program 95.1%

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

   2. Taylor expanded in y around 0 95.2%

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

      1. distribute-lft-out 95.2%

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

      2. unpow2 95.2%

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

      3. unpow2 95.2%

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

   4. Simplified 95.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-18} \lor \neg \left(y \leq 1.65 \cdot 10^{-96}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \frac{y - z}{y} \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \end{array} \]

Alternative 5: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z \cdot z}{y}\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{\frac{y}{y + z}}{y - z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-18)
   (* 0.5 (+ (/ x (/ y x)) (- y (/ (* z z) y))))
   (if (<= (* z z) 4e+284)
     (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))
     (/ 1.0 (* 2.0 (/ (/ y (+ y z)) (- y z)))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-18) {
		tmp = 0.5 * ((x / (y / x)) + (y - ((z * z) / y)));
	} else if ((z * z) <= 4e+284) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = 1.0 / (2.0 * ((y / (y + z)) / (y - z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-18) then
        tmp = 0.5d0 * ((x / (y / x)) + (y - ((z * z) / y)))
    else if ((z * z) <= 4d+284) then
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    else
        tmp = 1.0d0 / (2.0d0 * ((y / (y + z)) / (y - z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-18) {
		tmp = 0.5 * ((x / (y / x)) + (y - ((z * z) / y)));
	} else if ((z * z) <= 4e+284) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = 1.0 / (2.0 * ((y / (y + z)) / (y - z)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-18:
		tmp = 0.5 * ((x / (y / x)) + (y - ((z * z) / y)))
	elif (z * z) <= 4e+284:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	else:
		tmp = 1.0 / (2.0 * ((y / (y + z)) / (y - z)))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-18)
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(y - Float64(Float64(z * z) / y))));
	elseif (Float64(z * z) <= 4e+284)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(y / Float64(y + z)) / Float64(y - z))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-18)
		tmp = 0.5 * ((x / (y / x)) + (y - ((z * z) / y)));
	elseif ((z * z) <= 4e+284)
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	else
		tmp = 1.0 / (2.0 * ((y / (y + z)) / (y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-18], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y - N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e+284], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.0000000000000001e-18

   1. Initial program 73.4%

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

   2. Taylor expanded in x around inf 89.9%

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

      1. +-commutative 89.9%

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

      2. distribute-lft-out 89.9%

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

      3. unpow2 89.9%

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

      4. associate-/l* 99.9%

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

      5. unpow2 99.9%

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

   4. Simplified 99.9%

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

   if 1.0000000000000001e-18 < (*.f64 z z) < 4.00000000000000032e284

   1. Initial program 90.1%

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

   2. Taylor expanded in y around 0 96.6%

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

      1. distribute-lft-out 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

   4. Simplified 96.6%

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

   if 4.00000000000000032e284 < (*.f64 z z)

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0 57.3%

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

      1. unpow2 57.3%

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

      2. unpow2 57.3%

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

      3. difference-of-squares 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-/l* 86.0%

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

      6. associate-*r/ 86.0%

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

   4. Simplified 86.0%

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

      1. clear-num 86.0%

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

      2. inv-pow 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. unpow-1 86.0%

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

      2. *-lft-identity 86.0%

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

      3. times-frac 86.0%

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

      4. metadata-eval 86.0%

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

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{\frac{y}{y + z}}{y - z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-18}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z \cdot z}{y}\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{\frac{y}{y + z}}{y - z}}\\ \end{array} \]

Alternative 6: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+295}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+295)
   (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))
   (* 0.5 (+ y (/ x (/ y x))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+295) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+295) then
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    else
        tmp = 0.5d0 * (y + (x / (y / x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+295) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+295:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	else:
		tmp = 0.5 * (y + (x / (y / x)))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+295)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+295)
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	else
		tmp = 0.5 * (y + (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+295], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e295

   1. Initial program 78.6%

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

   2. Taylor expanded in y around 0 91.4%

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

      1. distribute-lft-out 91.4%

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

      2. unpow2 91.4%

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

      3. unpow2 91.4%

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

   4. Simplified 91.4%

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

   if 2e295 < (*.f64 x x)

   1. Initial program 50.8%

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

   2. Taylor expanded in y around 0 52.4%

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

      1. distribute-lft-out 52.4%

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

      2. unpow2 52.4%

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

      3. unpow2 52.4%

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

   4. Simplified 52.4%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. unpow2 66.7%

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

      2. associate-/l* 92.2%

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

   7. Simplified 92.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+295}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 7: 44.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := \left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{-200}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-6}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z (/ z y)) -0.5)))
   (if (<= x 3.2e-200)
     (* y 0.5)
     (if (<= x 7.2e-35)
       t_0
       (if (<= x 9.6e-6)
         (* y 0.5)
         (if (<= x 8.5e+50) t_0 (* 0.5 (/ x (/ y x)))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double tmp;
	if (x <= 3.2e-200) {
		tmp = y * 0.5;
	} else if (x <= 7.2e-35) {
		tmp = t_0;
	} else if (x <= 9.6e-6) {
		tmp = y * 0.5;
	} else if (x <= 8.5e+50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 = (z * (z / y)) * (-0.5d0)
    if (x <= 3.2d-200) then
        tmp = y * 0.5d0
    else if (x <= 7.2d-35) then
        tmp = t_0
    else if (x <= 9.6d-6) then
        tmp = y * 0.5d0
    else if (x <= 8.5d+50) then
        tmp = t_0
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double tmp;
	if (x <= 3.2e-200) {
		tmp = y * 0.5;
	} else if (x <= 7.2e-35) {
		tmp = t_0;
	} else if (x <= 9.6e-6) {
		tmp = y * 0.5;
	} else if (x <= 8.5e+50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = (z * (z / y)) * -0.5
	tmp = 0
	if x <= 3.2e-200:
		tmp = y * 0.5
	elif x <= 7.2e-35:
		tmp = t_0
	elif x <= 9.6e-6:
		tmp = y * 0.5
	elif x <= 8.5e+50:
		tmp = t_0
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(Float64(z * Float64(z / y)) * -0.5)
	tmp = 0.0
	if (x <= 3.2e-200)
		tmp = Float64(y * 0.5);
	elseif (x <= 7.2e-35)
		tmp = t_0;
	elseif (x <= 9.6e-6)
		tmp = Float64(y * 0.5);
	elseif (x <= 8.5e+50)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = (z * (z / y)) * -0.5;
	tmp = 0.0;
	if (x <= 3.2e-200)
		tmp = y * 0.5;
	elseif (x <= 7.2e-35)
		tmp = t_0;
	elseif (x <= 9.6e-6)
		tmp = y * 0.5;
	elseif (x <= 8.5e+50)
		tmp = t_0;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[x, 3.2e-200], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 7.2e-35], t$95$0, If[LessEqual[x, 9.6e-6], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 8.5e+50], t$95$0, N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.19999999999999983e-200 or 7.20000000000000038e-35 < x < 9.5999999999999996e-6

   1. Initial program 72.2%

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

   2. Taylor expanded in y around inf 36.1%

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

   if 3.19999999999999983e-200 < x < 7.20000000000000038e-35 or 9.5999999999999996e-6 < x < 8.49999999999999961e50

   1. Initial program 83.5%

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

   2. Taylor expanded in z around inf 57.5%

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

      1. *-commutative 57.5%

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

      2. unpow2 57.5%

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

   4. Simplified 57.5%

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

   5. Taylor expanded in z around 0 57.5%

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

      1. unpow2 57.5%

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

      2. associate-*r/ 59.8%

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

   7. Simplified 59.8%

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

   if 8.49999999999999961e50 < x

   1. Initial program 62.9%

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

   2. Taylor expanded in x around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-/l* 68.3%

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

   4. Simplified 68.3%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-200}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-35}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-6}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+50}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 8: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := \frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{if}\;x \leq 3 \cdot 10^{-208}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (* (/ y z) -2.0))))
   (if (<= x 3e-208)
     (* y 0.5)
     (if (<= x 5.8e-35)
       t_0
       (if (<= x 3.8e-7)
         (* y 0.5)
         (if (<= x 7e+50) t_0 (* 0.5 (/ x (/ y x)))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = z / ((y / z) * -2.0);
	double tmp;
	if (x <= 3e-208) {
		tmp = y * 0.5;
	} else if (x <= 5.8e-35) {
		tmp = t_0;
	} else if (x <= 3.8e-7) {
		tmp = y * 0.5;
	} else if (x <= 7e+50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 = z / ((y / z) * (-2.0d0))
    if (x <= 3d-208) then
        tmp = y * 0.5d0
    else if (x <= 5.8d-35) then
        tmp = t_0
    else if (x <= 3.8d-7) then
        tmp = y * 0.5d0
    else if (x <= 7d+50) then
        tmp = t_0
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double t_0 = z / ((y / z) * -2.0);
	double tmp;
	if (x <= 3e-208) {
		tmp = y * 0.5;
	} else if (x <= 5.8e-35) {
		tmp = t_0;
	} else if (x <= 3.8e-7) {
		tmp = y * 0.5;
	} else if (x <= 7e+50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = z / ((y / z) * -2.0)
	tmp = 0
	if x <= 3e-208:
		tmp = y * 0.5
	elif x <= 5.8e-35:
		tmp = t_0
	elif x <= 3.8e-7:
		tmp = y * 0.5
	elif x <= 7e+50:
		tmp = t_0
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(z / Float64(Float64(y / z) * -2.0))
	tmp = 0.0
	if (x <= 3e-208)
		tmp = Float64(y * 0.5);
	elseif (x <= 5.8e-35)
		tmp = t_0;
	elseif (x <= 3.8e-7)
		tmp = Float64(y * 0.5);
	elseif (x <= 7e+50)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = z / ((y / z) * -2.0);
	tmp = 0.0;
	if (x <= 3e-208)
		tmp = y * 0.5;
	elseif (x <= 5.8e-35)
		tmp = t_0;
	elseif (x <= 3.8e-7)
		tmp = y * 0.5;
	elseif (x <= 7e+50)
		tmp = t_0;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(N[(y / z), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3e-208], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 5.8e-35], t$95$0, If[LessEqual[x, 3.8e-7], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 7e+50], t$95$0, N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.99999999999999986e-208 or 5.8000000000000004e-35 < x < 3.80000000000000015e-7

   1. Initial program 72.2%

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

   2. Taylor expanded in y around inf 36.1%

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

   if 2.99999999999999986e-208 < x < 5.8000000000000004e-35 or 3.80000000000000015e-7 < x < 7.00000000000000012e50

   1. Initial program 83.5%

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

      1. div-sub 81.0%

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

      2. sub-neg 81.0%

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

      3. add-sqr-sqrt 81.0%

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

      4. times-frac 81.0%

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

      5. fma-def 83.5%

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

      6. hypot-def 83.5%

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

      7. hypot-def 97.5%

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

      8. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around inf 57.5%

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

      1. *-commutative 57.5%

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

      2. unpow2 57.5%

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

      3. metadata-eval 57.5%

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

      4. associate-/l* 59.9%

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

      5. times-frac 59.9%

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

      6. *-rgt-identity 59.9%

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

   6. Simplified 59.9%

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

   if 7.00000000000000012e50 < x

   1. Initial program 62.9%

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

   2. Taylor expanded in x around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-/l* 68.3%

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

   4. Simplified 68.3%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-208}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+50}:\\ \;\;\;\;\frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := \frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{if}\;x \leq 6.6 \cdot 10^{-205}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (* (/ y z) -2.0))))
   (if (<= x 6.6e-205)
     (* y 0.5)
     (if (<= x 4.2e-35)
       t_0
       (if (<= x 1.6e-8)
         (* y 0.5)
         (if (<= x 1.1e+51) t_0 (/ (* x (/ x y)) 2.0)))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = z / ((y / z) * -2.0);
	double tmp;
	if (x <= 6.6e-205) {
		tmp = y * 0.5;
	} else if (x <= 4.2e-35) {
		tmp = t_0;
	} else if (x <= 1.6e-8) {
		tmp = y * 0.5;
	} else if (x <= 1.1e+51) {
		tmp = t_0;
	} else {
		tmp = (x * (x / y)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 = z / ((y / z) * (-2.0d0))
    if (x <= 6.6d-205) then
        tmp = y * 0.5d0
    else if (x <= 4.2d-35) then
        tmp = t_0
    else if (x <= 1.6d-8) then
        tmp = y * 0.5d0
    else if (x <= 1.1d+51) then
        tmp = t_0
    else
        tmp = (x * (x / y)) / 2.0d0
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double t_0 = z / ((y / z) * -2.0);
	double tmp;
	if (x <= 6.6e-205) {
		tmp = y * 0.5;
	} else if (x <= 4.2e-35) {
		tmp = t_0;
	} else if (x <= 1.6e-8) {
		tmp = y * 0.5;
	} else if (x <= 1.1e+51) {
		tmp = t_0;
	} else {
		tmp = (x * (x / y)) / 2.0;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = z / ((y / z) * -2.0)
	tmp = 0
	if x <= 6.6e-205:
		tmp = y * 0.5
	elif x <= 4.2e-35:
		tmp = t_0
	elif x <= 1.6e-8:
		tmp = y * 0.5
	elif x <= 1.1e+51:
		tmp = t_0
	else:
		tmp = (x * (x / y)) / 2.0
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(z / Float64(Float64(y / z) * -2.0))
	tmp = 0.0
	if (x <= 6.6e-205)
		tmp = Float64(y * 0.5);
	elseif (x <= 4.2e-35)
		tmp = t_0;
	elseif (x <= 1.6e-8)
		tmp = Float64(y * 0.5);
	elseif (x <= 1.1e+51)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(x / y)) / 2.0);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = z / ((y / z) * -2.0);
	tmp = 0.0;
	if (x <= 6.6e-205)
		tmp = y * 0.5;
	elseif (x <= 4.2e-35)
		tmp = t_0;
	elseif (x <= 1.6e-8)
		tmp = y * 0.5;
	elseif (x <= 1.1e+51)
		tmp = t_0;
	else
		tmp = (x * (x / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(N[(y / z), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.6e-205], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 4.2e-35], t$95$0, If[LessEqual[x, 1.6e-8], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 1.1e+51], t$95$0, N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.5999999999999998e-205 or 4.2e-35 < x < 1.6000000000000001e-8

   1. Initial program 72.2%

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

   2. Taylor expanded in y around inf 36.1%

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

   if 6.5999999999999998e-205 < x < 4.2e-35 or 1.6000000000000001e-8 < x < 1.09999999999999996e51

   1. Initial program 83.5%

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

      1. div-sub 81.0%

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

      2. sub-neg 81.0%

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

      3. add-sqr-sqrt 81.0%

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

      4. times-frac 81.0%

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

      5. fma-def 83.5%

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

      6. hypot-def 83.5%

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

      7. hypot-def 97.5%

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

      8. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around inf 57.5%

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

      1. *-commutative 57.5%

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

      2. unpow2 57.5%

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

      3. metadata-eval 57.5%

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

      4. associate-/l* 59.9%

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

      5. times-frac 59.9%

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

      6. *-rgt-identity 59.9%

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

   6. Simplified 59.9%

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

   if 1.09999999999999996e51 < x

   1. Initial program 62.9%

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

   2. Taylor expanded in x around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-/l* 68.3%

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

   4. Simplified 68.3%

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

      1. metadata-eval 68.3%

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

      2. associate-/r/ 68.4%

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

      3. associate-*l/ 61.7%

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

      4. times-frac 61.7%

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

      5. *-un-lft-identity 61.7%

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

      6. *-commutative 61.7%

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

      7. associate-/r* 61.7%

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

      8. associate-*l/ 68.4%

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

      9. associate-/r/ 68.3%

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

      10. div-inv 68.3%

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

      11. clear-num 68.4%

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

   6. Applied egg-rr 68.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-205}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{z}{\frac{y}{z} \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \]

Alternative 10: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+284) (* 0.5 (+ y (/ x (/ y x)))) (* (* z (/ z y)) -0.5)))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+284) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 4d+284) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = (z * (z / y)) * (-0.5d0)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+284) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+284:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+284)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e+284)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+284], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000032e284

   1. Initial program 78.9%

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

   2. Taylor expanded in y around 0 92.1%

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

      1. distribute-lft-out 92.1%

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

      2. unpow2 92.1%

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

      3. unpow2 92.1%

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

   4. Simplified 92.1%

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

   5. Taylor expanded in x around inf 74.9%

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

      1. unpow2 74.9%

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

      2. associate-/l* 82.7%

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

   7. Simplified 82.7%

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

   if 4.00000000000000032e284 < (*.f64 z z)

   1. Initial program 54.7%

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

   2. Taylor expanded in z around inf 61.2%

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

      1. *-commutative 61.2%

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

      2. unpow2 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in z around 0 61.2%

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

      1. unpow2 61.2%

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

      2. associate-*r/ 71.8%

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

   7. Simplified 71.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 11: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+132)
   (* 0.5 (- y (/ z (/ y z))))
   (* 0.5 (+ y (/ x (/ y x))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+132) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+132) then
        tmp = 0.5d0 * (y - (z / (y / z)))
    else
        tmp = 0.5d0 * (y + (x / (y / x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+132) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (y + (x / (y / x)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+132:
		tmp = 0.5 * (y - (z / (y / z)))
	else:
		tmp = 0.5 * (y + (x / (y / x)))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+132)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+132)
		tmp = 0.5 * (y - (z / (y / z)));
	else
		tmp = 0.5 * (y + (x / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+132], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999998e132

   1. Initial program 77.7%

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

      1. div-sub 75.7%

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

      2. sub-neg 75.7%

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

      3. add-sqr-sqrt 75.7%

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

      4. times-frac 75.7%

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

      5. fma-def 77.7%

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

      6. hypot-def 77.7%

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

      7. hypot-def 93.5%

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

      8. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 85.0%

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

      1. distribute-lft-out-- 85.0%

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

      2. unpow2 85.0%

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

      3. associate-/l* 91.3%

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

   6. Simplified 91.3%

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

   if 1.99999999999999998e132 < (*.f64 x x)

   1. Initial program 62.9%

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

   2. Taylor expanded in y around 0 65.6%

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

      1. distribute-lft-out 65.6%

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

      2. unpow2 65.6%

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

      3. unpow2 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in x around inf 74.1%

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

      1. unpow2 74.1%

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

      2. associate-/l* 89.6%

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

   7. Simplified 89.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 12: 44.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.65e+66) (* y 0.5) (* 0.5 (/ x (/ y x)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.65e+66) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.65d+66) then
        tmp = y * 0.5d0
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.65e+66) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 1.65e+66:
		tmp = y * 0.5
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.65e+66)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.65e+66)
		tmp = y * 0.5;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1.65e+66], N[(y * 0.5), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6500000000000001e66

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf 34.9%

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

   if 1.6500000000000001e66 < x

   1. Initial program 62.1%

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

   2. Taylor expanded in x around inf 63.8%

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

      1. unpow2 63.8%

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

      2. associate-/l* 70.9%

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

   4. Simplified 70.9%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 13: 34.8% accurate, 5.0× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z) :precision binary64 (* y 0.5))

C

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

Fortran

NOTE: z should be positive before calling this function
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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

2. Taylor expanded in y around inf 31.2%

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

3. Final simplification 31.2%

   \[\leadsto y \cdot 0.5 \]

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 2023274 
(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)))