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

Percentage Accurate: 69.3% → 96.9%

Time: 8.8s

Alternatives: 14

Speedup: 1.4×

• 41.8% 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: 69.3% 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: 96.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-133} \lor \neg \left(y \leq 3 \cdot 10^{-230}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{y}, \frac{\mathsf{hypot}\left(x, y\right)}{2}, \frac{z}{y} \cdot \frac{-z}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6e-133) (not (<= y 3e-230)))
   (fma (/ (hypot x y) y) (/ (hypot x y) 2.0) (* (/ z y) (/ (- z) 2.0)))
   (/ 1.0 (* (/ y (- x z)) (/ 2.0 (+ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6e-133) || !(y <= 3e-230)) {
		tmp = fma((hypot(x, y) / y), (hypot(x, y) / 2.0), ((z / y) * (-z / 2.0)));
	} else {
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6e-133) || !(y <= 3e-230))
		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(1.0 / Float64(Float64(y / Float64(x - z)) * Float64(2.0 / Float64(x + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6e-133], N[Not[LessEqual[y, 3e-230]], $MachinePrecision]], 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[(1.0 / N[(N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000038e-133 or 3e-230 < y

   1. Initial program 62.4%

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

      1. div-sub 60.4%

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

      2. sub-neg 60.4%

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

      3. add-sqr-sqrt 60.4%

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

      4. times-frac 60.5%

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

      5. fma-def 62.5%

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

      6. hypot-def 62.5%

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

      7. hypot-def 88.2%

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

      8. times-frac 98.4%

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

   3. Applied egg-rr 98.4%

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

   if -6.00000000000000038e-133 < y < 3e-230

   1. Initial program 93.1%

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

   2. Taylor expanded in y around 0 91.2%

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

      1. unpow2 91.2%

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

      2. unpow2 91.2%

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

   4. Simplified 91.2%

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

      1. clear-num 91.2%

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

      2. inv-pow 91.2%

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

      3. *-commutative 91.2%

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

      4. *-un-lft-identity 91.2%

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

      5. times-frac 91.2%

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

      6. metadata-eval 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. unpow-1 91.2%

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

      2. associate-*r/ 91.2%

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

      3. *-commutative 91.2%

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

      4. difference-of-squares 93.2%

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

      5. *-commutative 93.2%

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

      6. times-frac 96.1%

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

      7. +-commutative 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-133} \lor \neg \left(y \leq 3 \cdot 10^{-230}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{hypot}\left(x, y\right)}{y}, \frac{\mathsf{hypot}\left(x, y\right)}{2}, \frac{z}{y} \cdot \frac{-z}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \]

Alternative 2: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+45} \lor \neg \left(y \leq 2.2 \cdot 10^{-217}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.18e+45) (not (<= y 2.2e-217)))
   (* 0.5 (+ (/ x (/ y x)) (- y (/ z (/ y z)))))
   (/ (fma x x (* (+ y z) (- y z))) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.18e+45) || !(y <= 2.2e-217)) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else {
		tmp = fma(x, x, ((y + z) * (y - z))) / (y * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.18e+45) || !(y <= 2.2e-217))
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(y - Float64(z / 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

code[x_, y_, z_] := If[Or[LessEqual[y, -1.18e+45], N[Not[LessEqual[y, 2.2e-217]], $MachinePrecision]], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y - N[(z / N[(y / z), $MachinePrecision]), $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 < -1.17999999999999993e45 or 2.19999999999999982e-217 < y

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf 75.1%

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

      1. +-commutative 75.1%

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

      2. distribute-lft-out 75.1%

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

      3. unpow2 75.1%

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

      4. associate-/l* 82.9%

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

      5. unpow2 82.9%

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

      6. associate-/l* 95.7%

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

   4. Simplified 95.7%

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

   if -1.17999999999999993e45 < y < 2.19999999999999982e-217

   1. Initial program 95.0%

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

      1. associate--l+ 95.0%

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

      2. fma-def 97.2%

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

      3. difference-of-squares 97.2%

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

   3. Simplified 97.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+45} \lor \neg \left(y \leq 2.2 \cdot 10^{-217}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y \cdot 2}\\ \end{array} \]

Alternative 3: 52.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{+196}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+250}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))))
   (if (<= (* z z) 5e+17)
     (* y 0.5)
     (if (<= (* z z) 1e+73)
       t_0
       (if (<= (* z z) 1e+196)
         (* z (* z (/ -0.5 y)))
         (if (<= (* z z) 1e+218)
           t_0
           (if (<= (* z z) 5e+250)
             (* y 0.5)
             (if (<= (* z z) 2e+256)
               (* (/ x y) (* x 0.5))
               (* (/ z y) (* z -0.5))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if ((z * z) <= 5e+17) {
		tmp = y * 0.5;
	} else if ((z * z) <= 1e+73) {
		tmp = t_0;
	} else if ((z * z) <= 1e+196) {
		tmp = z * (z * (-0.5 / y));
	} else if ((z * z) <= 1e+218) {
		tmp = t_0;
	} else if ((z * z) <= 5e+250) {
		tmp = y * 0.5;
	} else if ((z * z) <= 2e+256) {
		tmp = (x / y) * (x * 0.5);
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / (y / x))
    if ((z * z) <= 5d+17) then
        tmp = y * 0.5d0
    else if ((z * z) <= 1d+73) then
        tmp = t_0
    else if ((z * z) <= 1d+196) then
        tmp = z * (z * ((-0.5d0) / y))
    else if ((z * z) <= 1d+218) then
        tmp = t_0
    else if ((z * z) <= 5d+250) then
        tmp = y * 0.5d0
    else if ((z * z) <= 2d+256) then
        tmp = (x / y) * (x * 0.5d0)
    else
        tmp = (z / y) * (z * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if ((z * z) <= 5e+17) {
		tmp = y * 0.5;
	} else if ((z * z) <= 1e+73) {
		tmp = t_0;
	} else if ((z * z) <= 1e+196) {
		tmp = z * (z * (-0.5 / y));
	} else if ((z * z) <= 1e+218) {
		tmp = t_0;
	} else if ((z * z) <= 5e+250) {
		tmp = y * 0.5;
	} else if ((z * z) <= 2e+256) {
		tmp = (x / y) * (x * 0.5);
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	tmp = 0
	if (z * z) <= 5e+17:
		tmp = y * 0.5
	elif (z * z) <= 1e+73:
		tmp = t_0
	elif (z * z) <= 1e+196:
		tmp = z * (z * (-0.5 / y))
	elif (z * z) <= 1e+218:
		tmp = t_0
	elif (z * z) <= 5e+250:
		tmp = y * 0.5
	elif (z * z) <= 2e+256:
		tmp = (x / y) * (x * 0.5)
	else:
		tmp = (z / y) * (z * -0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (Float64(z * z) <= 5e+17)
		tmp = Float64(y * 0.5);
	elseif (Float64(z * z) <= 1e+73)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e+196)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	elseif (Float64(z * z) <= 1e+218)
		tmp = t_0;
	elseif (Float64(z * z) <= 5e+250)
		tmp = Float64(y * 0.5);
	elseif (Float64(z * z) <= 2e+256)
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	else
		tmp = Float64(Float64(z / y) * Float64(z * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if ((z * z) <= 5e+17)
		tmp = y * 0.5;
	elseif ((z * z) <= 1e+73)
		tmp = t_0;
	elseif ((z * z) <= 1e+196)
		tmp = z * (z * (-0.5 / y));
	elseif ((z * z) <= 1e+218)
		tmp = t_0;
	elseif ((z * z) <= 5e+250)
		tmp = y * 0.5;
	elseif ((z * z) <= 2e+256)
		tmp = (x / y) * (x * 0.5);
	else
		tmp = (z / y) * (z * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e+17], N[(y * 0.5), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+73], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e+196], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+218], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 5e+250], N[(y * 0.5), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+256], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z z) < 5e17 or 1.00000000000000008e218 < (*.f64 z z) < 5.0000000000000002e250

   1. Initial program 70.8%

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

   2. Taylor expanded in y around inf 62.1%

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

   if 5e17 < (*.f64 z z) < 9.99999999999999983e72 or 9.9999999999999995e195 < (*.f64 z z) < 1.00000000000000008e218

   1. Initial program 94.1%

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

   2. Taylor expanded in x around inf 73.8%

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

      1. unpow2 73.8%

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

      2. associate-/l* 73.8%

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

   4. Simplified 73.8%

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

   if 9.99999999999999983e72 < (*.f64 z z) < 9.9999999999999995e195

   1. Initial program 87.8%

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

      1. div-sub 66.1%

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

      2. sub-neg 66.1%

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

      3. add-sqr-sqrt 66.1%

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

      4. times-frac 66.1%

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

      5. fma-def 66.1%

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

      6. hypot-def 66.1%

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

      7. hypot-def 78.1%

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

      8. times-frac 78.1%

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

   3. Applied egg-rr 78.1%

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

   4. Taylor expanded in z around inf 52.6%

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

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. associate-*l/ 52.6%

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

      4. associate-*l* 52.6%

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

   6. Simplified 52.6%

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

      1. associate-*l/ 52.6%

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

      2. associate-*r* 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. div-inv 52.6%

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

      2. associate-*l* 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-*l* 52.7%

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

      5. associate-*l/ 52.7%

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

      6. metadata-eval 52.7%

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

   10. Applied egg-rr 52.7%

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

   if 5.0000000000000002e250 < (*.f64 z z) < 2.0000000000000001e256

   1. Initial program 77.0%

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

   2. Taylor expanded in x around inf 77.0%

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

      1. unpow2 77.0%

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

   4. Simplified 77.0%

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

      1. times-frac 100.0%

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

      2. div-inv 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.0000000000000001e256 < (*.f64 z z)

   1. Initial program 51.1%

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

      1. div-sub 42.7%

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

      2. sub-neg 42.7%

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

      3. add-sqr-sqrt 42.7%

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

      4. times-frac 42.7%

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

      5. fma-def 48.3%

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

      6. hypot-def 48.3%

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

      7. hypot-def 61.7%

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

      8. times-frac 91.5%

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

   3. Applied egg-rr 91.5%

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

   4. Taylor expanded in z around inf 61.8%

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

      1. *-commutative 61.8%

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

      2. unpow2 61.8%

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

      3. associate-*l/ 74.5%

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

      4. associate-*l* 74.5%

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

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{z}{y} \cdot \left(z \cdot -0.5\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 10^{+73}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 10^{+196}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+218}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+250}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \]

Alternative 4: 44.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ t_1 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;z \leq 4.4 \cdot 10^{-272}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-181}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1300000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+125}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ -0.5 y)))) (t_1 (* 0.5 (/ x (/ y x)))))
   (if (<= z 4.4e-272)
     (* y 0.5)
     (if (<= z 4.8e-226)
       t_1
       (if (<= z 1e-181)
         (* y 0.5)
         (if (<= z 2e-155)
           t_1
           (if (<= z 1300000000000.0)
             (* y 0.5)
             (if (<= z 2.8e+36)
               t_1
               (if (<= z 5.6e+99)
                 t_0
                 (if (<= z 1.3e+109)
                   t_1
                   (if (<= z 5.4e+125)
                     (* y 0.5)
                     (if (<= z 1.35e+128) t_1 t_0))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 4.4e-272) {
		tmp = y * 0.5;
	} else if (z <= 4.8e-226) {
		tmp = t_1;
	} else if (z <= 1e-181) {
		tmp = y * 0.5;
	} else if (z <= 2e-155) {
		tmp = t_1;
	} else if (z <= 1300000000000.0) {
		tmp = y * 0.5;
	} else if (z <= 2.8e+36) {
		tmp = t_1;
	} else if (z <= 5.6e+99) {
		tmp = t_0;
	} else if (z <= 1.3e+109) {
		tmp = t_1;
	} else if (z <= 5.4e+125) {
		tmp = y * 0.5;
	} else if (z <= 1.35e+128) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (z * ((-0.5d0) / y))
    t_1 = 0.5d0 * (x / (y / x))
    if (z <= 4.4d-272) then
        tmp = y * 0.5d0
    else if (z <= 4.8d-226) then
        tmp = t_1
    else if (z <= 1d-181) then
        tmp = y * 0.5d0
    else if (z <= 2d-155) then
        tmp = t_1
    else if (z <= 1300000000000.0d0) then
        tmp = y * 0.5d0
    else if (z <= 2.8d+36) then
        tmp = t_1
    else if (z <= 5.6d+99) then
        tmp = t_0
    else if (z <= 1.3d+109) then
        tmp = t_1
    else if (z <= 5.4d+125) then
        tmp = y * 0.5d0
    else if (z <= 1.35d+128) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 4.4e-272) {
		tmp = y * 0.5;
	} else if (z <= 4.8e-226) {
		tmp = t_1;
	} else if (z <= 1e-181) {
		tmp = y * 0.5;
	} else if (z <= 2e-155) {
		tmp = t_1;
	} else if (z <= 1300000000000.0) {
		tmp = y * 0.5;
	} else if (z <= 2.8e+36) {
		tmp = t_1;
	} else if (z <= 5.6e+99) {
		tmp = t_0;
	} else if (z <= 1.3e+109) {
		tmp = t_1;
	} else if (z <= 5.4e+125) {
		tmp = y * 0.5;
	} else if (z <= 1.35e+128) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * (-0.5 / y))
	t_1 = 0.5 * (x / (y / x))
	tmp = 0
	if z <= 4.4e-272:
		tmp = y * 0.5
	elif z <= 4.8e-226:
		tmp = t_1
	elif z <= 1e-181:
		tmp = y * 0.5
	elif z <= 2e-155:
		tmp = t_1
	elif z <= 1300000000000.0:
		tmp = y * 0.5
	elif z <= 2.8e+36:
		tmp = t_1
	elif z <= 5.6e+99:
		tmp = t_0
	elif z <= 1.3e+109:
		tmp = t_1
	elif z <= 5.4e+125:
		tmp = y * 0.5
	elif z <= 1.35e+128:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * Float64(-0.5 / y)))
	t_1 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (z <= 4.4e-272)
		tmp = Float64(y * 0.5);
	elseif (z <= 4.8e-226)
		tmp = t_1;
	elseif (z <= 1e-181)
		tmp = Float64(y * 0.5);
	elseif (z <= 2e-155)
		tmp = t_1;
	elseif (z <= 1300000000000.0)
		tmp = Float64(y * 0.5);
	elseif (z <= 2.8e+36)
		tmp = t_1;
	elseif (z <= 5.6e+99)
		tmp = t_0;
	elseif (z <= 1.3e+109)
		tmp = t_1;
	elseif (z <= 5.4e+125)
		tmp = Float64(y * 0.5);
	elseif (z <= 1.35e+128)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * (-0.5 / y));
	t_1 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if (z <= 4.4e-272)
		tmp = y * 0.5;
	elseif (z <= 4.8e-226)
		tmp = t_1;
	elseif (z <= 1e-181)
		tmp = y * 0.5;
	elseif (z <= 2e-155)
		tmp = t_1;
	elseif (z <= 1300000000000.0)
		tmp = y * 0.5;
	elseif (z <= 2.8e+36)
		tmp = t_1;
	elseif (z <= 5.6e+99)
		tmp = t_0;
	elseif (z <= 1.3e+109)
		tmp = t_1;
	elseif (z <= 5.4e+125)
		tmp = y * 0.5;
	elseif (z <= 1.35e+128)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 4.4e-272], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 4.8e-226], t$95$1, If[LessEqual[z, 1e-181], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 2e-155], t$95$1, If[LessEqual[z, 1300000000000.0], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 2.8e+36], t$95$1, If[LessEqual[z, 5.6e+99], t$95$0, If[LessEqual[z, 1.3e+109], t$95$1, If[LessEqual[z, 5.4e+125], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 1.35e+128], t$95$1, t$95$0]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.39999999999999976e-272 or 4.7999999999999999e-226 < z < 1.00000000000000005e-181 or 2.00000000000000003e-155 < z < 1.3e12 or 1.2999999999999999e109 < z < 5.3999999999999997e125

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 45.6%

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

   if 4.39999999999999976e-272 < z < 4.7999999999999999e-226 or 1.00000000000000005e-181 < z < 2.00000000000000003e-155 or 1.3e12 < z < 2.8000000000000001e36 or 5.6e99 < z < 1.2999999999999999e109 or 5.3999999999999997e125 < z < 1.35000000000000001e128

   1. Initial program 74.1%

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

   2. Taylor expanded in x around inf 57.3%

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

      1. unpow2 57.3%

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

      2. associate-/l* 62.3%

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

   4. Simplified 62.3%

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

   if 2.8000000000000001e36 < z < 5.6e99 or 1.35000000000000001e128 < z

   1. Initial program 62.9%

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

      1. div-sub 49.4%

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

      2. sub-neg 49.4%

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

      3. add-sqr-sqrt 49.4%

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

      4. times-frac 49.4%

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

      5. fma-def 54.8%

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

      6. hypot-def 54.8%

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

      7. hypot-def 71.3%

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

      8. times-frac 91.8%

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

   3. Applied egg-rr 91.8%

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

   4. Taylor expanded in z around inf 60.6%

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

      1. *-commutative 60.6%

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

      2. unpow2 60.6%

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

      3. associate-*l/ 73.0%

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

      4. associate-*l* 73.0%

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

   6. Simplified 73.0%

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

      1. associate-*l/ 60.6%

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

      2. associate-*r* 60.6%

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

   8. Applied egg-rr 60.6%

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

      1. div-inv 60.6%

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

      2. associate-*l* 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*l* 73.0%

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

      5. associate-*l/ 73.0%

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

      6. metadata-eval 73.0%

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

   10. Applied egg-rr 73.0%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-272}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 10^{-181}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1300000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+99}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+125}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]

Alternative 5: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ t_1 := \frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ t_2 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;z \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1300000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ -0.5 y))))
        (t_1 (* (/ x y) (* x 0.5)))
        (t_2 (* 0.5 (/ x (/ y x)))))
   (if (<= z 9.2e-272)
     (* y 0.5)
     (if (<= z 5.4e-226)
       t_2
       (if (<= z 1.6e-180)
         (* y 0.5)
         (if (<= z 6.5e-155)
           t_1
           (if (<= z 1300000000000.0)
             (* y 0.5)
             (if (<= z 1.9e+37)
               t_2
               (if (<= z 5.6e+99)
                 t_0
                 (if (<= z 1.8e+109)
                   t_2
                   (if (<= z 1.7e+126)
                     (* y 0.5)
                     (if (<= z 1.35e+128) t_1 t_0))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = (x / y) * (x * 0.5);
	double t_2 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 9.2e-272) {
		tmp = y * 0.5;
	} else if (z <= 5.4e-226) {
		tmp = t_2;
	} else if (z <= 1.6e-180) {
		tmp = y * 0.5;
	} else if (z <= 6.5e-155) {
		tmp = t_1;
	} else if (z <= 1300000000000.0) {
		tmp = y * 0.5;
	} else if (z <= 1.9e+37) {
		tmp = t_2;
	} else if (z <= 5.6e+99) {
		tmp = t_0;
	} else if (z <= 1.8e+109) {
		tmp = t_2;
	} else if (z <= 1.7e+126) {
		tmp = y * 0.5;
	} else if (z <= 1.35e+128) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z * (z * ((-0.5d0) / y))
    t_1 = (x / y) * (x * 0.5d0)
    t_2 = 0.5d0 * (x / (y / x))
    if (z <= 9.2d-272) then
        tmp = y * 0.5d0
    else if (z <= 5.4d-226) then
        tmp = t_2
    else if (z <= 1.6d-180) then
        tmp = y * 0.5d0
    else if (z <= 6.5d-155) then
        tmp = t_1
    else if (z <= 1300000000000.0d0) then
        tmp = y * 0.5d0
    else if (z <= 1.9d+37) then
        tmp = t_2
    else if (z <= 5.6d+99) then
        tmp = t_0
    else if (z <= 1.8d+109) then
        tmp = t_2
    else if (z <= 1.7d+126) then
        tmp = y * 0.5d0
    else if (z <= 1.35d+128) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = (x / y) * (x * 0.5);
	double t_2 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 9.2e-272) {
		tmp = y * 0.5;
	} else if (z <= 5.4e-226) {
		tmp = t_2;
	} else if (z <= 1.6e-180) {
		tmp = y * 0.5;
	} else if (z <= 6.5e-155) {
		tmp = t_1;
	} else if (z <= 1300000000000.0) {
		tmp = y * 0.5;
	} else if (z <= 1.9e+37) {
		tmp = t_2;
	} else if (z <= 5.6e+99) {
		tmp = t_0;
	} else if (z <= 1.8e+109) {
		tmp = t_2;
	} else if (z <= 1.7e+126) {
		tmp = y * 0.5;
	} else if (z <= 1.35e+128) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * (-0.5 / y))
	t_1 = (x / y) * (x * 0.5)
	t_2 = 0.5 * (x / (y / x))
	tmp = 0
	if z <= 9.2e-272:
		tmp = y * 0.5
	elif z <= 5.4e-226:
		tmp = t_2
	elif z <= 1.6e-180:
		tmp = y * 0.5
	elif z <= 6.5e-155:
		tmp = t_1
	elif z <= 1300000000000.0:
		tmp = y * 0.5
	elif z <= 1.9e+37:
		tmp = t_2
	elif z <= 5.6e+99:
		tmp = t_0
	elif z <= 1.8e+109:
		tmp = t_2
	elif z <= 1.7e+126:
		tmp = y * 0.5
	elif z <= 1.35e+128:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * Float64(-0.5 / y)))
	t_1 = Float64(Float64(x / y) * Float64(x * 0.5))
	t_2 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (z <= 9.2e-272)
		tmp = Float64(y * 0.5);
	elseif (z <= 5.4e-226)
		tmp = t_2;
	elseif (z <= 1.6e-180)
		tmp = Float64(y * 0.5);
	elseif (z <= 6.5e-155)
		tmp = t_1;
	elseif (z <= 1300000000000.0)
		tmp = Float64(y * 0.5);
	elseif (z <= 1.9e+37)
		tmp = t_2;
	elseif (z <= 5.6e+99)
		tmp = t_0;
	elseif (z <= 1.8e+109)
		tmp = t_2;
	elseif (z <= 1.7e+126)
		tmp = Float64(y * 0.5);
	elseif (z <= 1.35e+128)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * (-0.5 / y));
	t_1 = (x / y) * (x * 0.5);
	t_2 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if (z <= 9.2e-272)
		tmp = y * 0.5;
	elseif (z <= 5.4e-226)
		tmp = t_2;
	elseif (z <= 1.6e-180)
		tmp = y * 0.5;
	elseif (z <= 6.5e-155)
		tmp = t_1;
	elseif (z <= 1300000000000.0)
		tmp = y * 0.5;
	elseif (z <= 1.9e+37)
		tmp = t_2;
	elseif (z <= 5.6e+99)
		tmp = t_0;
	elseif (z <= 1.8e+109)
		tmp = t_2;
	elseif (z <= 1.7e+126)
		tmp = y * 0.5;
	elseif (z <= 1.35e+128)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 9.2e-272], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 5.4e-226], t$95$2, If[LessEqual[z, 1.6e-180], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 6.5e-155], t$95$1, If[LessEqual[z, 1300000000000.0], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 1.9e+37], t$95$2, If[LessEqual[z, 5.6e+99], t$95$0, If[LessEqual[z, 1.8e+109], t$95$2, If[LessEqual[z, 1.7e+126], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 1.35e+128], t$95$1, t$95$0]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 9.19999999999999955e-272 or 5.40000000000000029e-226 < z < 1.60000000000000008e-180 or 6.5e-155 < z < 1.3e12 or 1.8e109 < z < 1.69999999999999995e126

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 45.6%

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

   if 9.19999999999999955e-272 < z < 5.40000000000000029e-226 or 1.3e12 < z < 1.89999999999999995e37 or 5.6e99 < z < 1.8e109

   1. Initial program 83.4%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. unpow2 64.9%

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

      2. associate-/l* 64.9%

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

   4. Simplified 64.9%

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

   if 1.60000000000000008e-180 < z < 6.5e-155 or 1.69999999999999995e126 < z < 1.35000000000000001e128

   1. Initial program 59.7%

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

   2. Taylor expanded in x around inf 45.5%

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

      1. unpow2 45.5%

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

   4. Simplified 45.5%

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

      1. times-frac 58.8%

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

      2. div-inv 58.8%

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

      3. metadata-eval 58.8%

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

   6. Applied egg-rr 58.8%

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

   if 1.89999999999999995e37 < z < 5.6e99 or 1.35000000000000001e128 < z

   1. Initial program 62.9%

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

      1. div-sub 49.4%

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

      2. sub-neg 49.4%

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

      3. add-sqr-sqrt 49.4%

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

      4. times-frac 49.4%

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

      5. fma-def 54.8%

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

      6. hypot-def 54.8%

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

      7. hypot-def 71.3%

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

      8. times-frac 91.8%

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

   3. Applied egg-rr 91.8%

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

   4. Taylor expanded in z around inf 60.6%

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

      1. *-commutative 60.6%

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

      2. unpow2 60.6%

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

      3. associate-*l/ 73.0%

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

      4. associate-*l* 73.0%

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

   6. Simplified 73.0%

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

      1. associate-*l/ 60.6%

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

      2. associate-*r* 60.6%

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

   8. Applied egg-rr 60.6%

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

      1. div-inv 60.6%

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

      2. associate-*l* 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*l* 73.0%

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

      5. associate-*l/ 73.0%

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

      6. metadata-eval 73.0%

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

   10. Applied egg-rr 73.0%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;z \leq 1300000000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+99}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+128}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]

Alternative 6: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+43} \lor \neg \left(y \leq 6.4 \cdot 10^{-223}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e+43) (not (<= y 6.4e-223)))
   (* 0.5 (+ (/ x (/ y x)) (- y (/ z (/ y z)))))
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e+43) || !(y <= 6.4e-223)) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7d+43)) .or. (.not. (y <= 6.4d-223))) then
        tmp = 0.5d0 * ((x / (y / x)) + (y - (z / (y / z))))
    else
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e+43) || !(y <= 6.4e-223)) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7e+43) or not (y <= 6.4e-223):
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))))
	else:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7e+43) || !(y <= 6.4e-223))
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(y - Float64(z / Float64(y / z)))));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7e+43) || ~((y <= 6.4e-223)))
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	else
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7e+43], N[Not[LessEqual[y, 6.4e-223]], $MachinePrecision]], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000002e43 or 6.4000000000000001e-223 < y

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf 75.1%

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

      1. +-commutative 75.1%

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

      2. distribute-lft-out 75.1%

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

      3. unpow2 75.1%

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

      4. associate-/l* 82.9%

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

      5. unpow2 82.9%

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

      6. associate-/l* 95.7%

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

   4. Simplified 95.7%

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

   if -7.0000000000000002e43 < y < 6.4000000000000001e-223

   1. Initial program 95.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+43} \lor \neg \left(y \leq 6.4 \cdot 10^{-223}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 7: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e+263)
   (* 0.5 (+ (/ x (/ y x)) (- y (/ z (/ y z)))))
   (/ 1.0 (* (/ y (- x z)) (/ 2.0 (+ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+263) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else {
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d+263) then
        tmp = 0.5d0 * ((x / (y / x)) + (y - (z / (y / z))))
    else
        tmp = 1.0d0 / ((y / (x - z)) * (2.0d0 / (x + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+263) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else {
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e+263:
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))))
	else:
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+263)
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(y - Float64(z / Float64(y / z)))));
	else
		tmp = Float64(1.0 / Float64(Float64(y / Float64(x - z)) * Float64(2.0 / Float64(x + z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e+263)
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	else
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+263], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000022e263

   1. Initial program 72.1%

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

   2. Taylor expanded in x around inf 88.6%

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

      1. +-commutative 88.6%

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

      2. distribute-lft-out 88.6%

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

      3. unpow2 88.6%

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

      4. associate-/l* 89.3%

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

      5. unpow2 89.3%

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

      6. associate-/l* 97.2%

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

   4. Simplified 97.2%

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

   if 5.00000000000000022e263 < (*.f64 x x)

   1. Initial program 57.7%

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

   2. Taylor expanded in y around 0 57.7%

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

      1. unpow2 57.7%

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

      2. unpow2 57.7%

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

   4. Simplified 57.7%

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

      1. clear-num 57.7%

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

      2. inv-pow 57.7%

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

      3. *-commutative 57.7%

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

      4. *-un-lft-identity 57.7%

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

      5. times-frac 57.7%

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

      6. metadata-eval 57.7%

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

   6. Applied egg-rr 57.7%

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

      1. unpow-1 57.7%

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

      2. associate-*r/ 57.7%

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

      3. *-commutative 57.7%

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

      4. difference-of-squares 77.5%

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

      5. *-commutative 77.5%

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

      6. times-frac 83.9%

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

      7. +-commutative 83.9%

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

   8. Simplified 83.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \]

Alternative 8: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4e+131)
   (* 0.5 (* (- y z) (/ (+ y z) y)))
   (/ 1.0 (* (/ y (- x z)) (/ 2.0 (+ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e+131) {
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	} else {
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 4d+131) then
        tmp = 0.5d0 * ((y - z) * ((y + z) / y))
    else
        tmp = 1.0d0 / ((y / (x - z)) * (2.0d0 / (x + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e+131) {
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	} else {
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 4e+131:
		tmp = 0.5 * ((y - z) * ((y + z) / y))
	else:
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4e+131)
		tmp = Float64(0.5 * Float64(Float64(y - z) * Float64(Float64(y + z) / y)));
	else
		tmp = Float64(1.0 / Float64(Float64(y / Float64(x - z)) * Float64(2.0 / Float64(x + z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 4e+131)
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	else
		tmp = 1.0 / ((y / (x - z)) * (2.0 / (x + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e+131], N[(0.5 * N[(N[(y - z), $MachinePrecision] * N[(N[(y + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.9999999999999996e131

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 64.0%

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

      1. unpow2 64.0%

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

      2. unpow2 64.0%

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

      3. difference-of-squares 65.8%

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

      4. associate-/l* 90.2%

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

      5. associate-/r/ 90.2%

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

   4. Simplified 90.2%

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

   if 3.9999999999999996e131 < (*.f64 x x)

   1. Initial program 61.6%

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

   2. Taylor expanded in y around 0 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

   4. Simplified 59.5%

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

      1. clear-num 59.5%

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

      2. inv-pow 59.5%

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

      3. *-commutative 59.5%

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

      4. *-un-lft-identity 59.5%

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

      5. times-frac 59.5%

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

      6. metadata-eval 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. unpow-1 59.5%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. difference-of-squares 72.5%

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

      5. *-commutative 72.5%

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

      6. times-frac 76.8%

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

      7. +-commutative 76.8%

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

   8. Simplified 76.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x - z} \cdot \frac{2}{x + z}}\\ \end{array} \]

Alternative 9: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-45}:\\ \;\;\;\;\frac{x \cdot x - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e-26)
   (* 0.5 (- y (* z (/ z y))))
   (if (<= y 8e-45)
     (/ (- (* x x) (* z z)) (* y 2.0))
     (* 0.5 (- y (/ z (/ y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-26) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else if (y <= 8e-45) {
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.8d-26)) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else if (y <= 8d-45) then
        tmp = ((x * x) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * (y - (z / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-26) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else if (y <= 8e-45) {
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e-26:
		tmp = 0.5 * (y - (z * (z / y)))
	elif y <= 8e-45:
		tmp = ((x * x) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * (y - (z / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e-26)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	elseif (y <= 8e-45)
		tmp = Float64(Float64(Float64(x * x) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e-26)
		tmp = 0.5 * (y - (z * (z / y)));
	elseif (y <= 8e-45)
		tmp = ((x * x) - (z * z)) / (y * 2.0);
	else
		tmp = 0.5 * (y - (z / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e-26], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-45], N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.80000000000000015e-26

   1. Initial program 45.3%

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

      1. div-sub 45.3%

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

      2. sub-neg 45.3%

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

      3. add-sqr-sqrt 45.3%

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

      4. times-frac 45.3%

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

      5. fma-def 45.3%

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

      6. hypot-def 45.3%

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

      7. hypot-def 82.1%

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

      8. times-frac 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around 0 71.6%

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

      1. distribute-lft-out-- 71.6%

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

      2. unpow2 71.6%

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

      3. associate-*r/ 83.4%

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

   6. Simplified 83.4%

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

   if -3.80000000000000015e-26 < y < 7.99999999999999987e-45

   1. Initial program 91.9%

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

   2. Taylor expanded in y around 0 84.6%

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

      1. unpow2 84.6%

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

      2. unpow2 84.6%

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

   4. Simplified 84.6%

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

   if 7.99999999999999987e-45 < y

   1. Initial program 60.9%

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

   2. Taylor expanded in x around inf 82.7%

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

      1. +-commutative 82.7%

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

      2. distribute-lft-out 82.7%

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

      3. unpow2 82.7%

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

      4. associate-/l* 88.0%

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

      5. unpow2 88.0%

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

      6. associate-/l* 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in x around 0 74.5%

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

      1. unpow2 74.5%

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

      2. associate-/l* 83.6%

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

   7. Simplified 83.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-45}:\\ \;\;\;\;\frac{x \cdot x - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 10: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.6e+163) (* 0.5 (* (- y z) (/ (+ y z) y))) (* (/ x y) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.6e+163) {
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.6d+163) then
        tmp = 0.5d0 * ((y - z) * ((y + z) / y))
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.6e+163) {
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.6e+163:
		tmp = 0.5 * ((y - z) * ((y + z) / y))
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.6e+163)
		tmp = Float64(0.5 * Float64(Float64(y - z) * Float64(Float64(y + z) / y)));
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.6e+163)
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.6e+163], N[(0.5 * N[(N[(y - z), $MachinePrecision] * N[(N[(y + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e163

   1. Initial program 69.7%

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

   2. Taylor expanded in x around 0 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. difference-of-squares 53.3%

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

      4. associate-/l* 75.6%

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

      5. associate-/r/ 75.6%

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

   4. Simplified 75.6%

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

   if 2.6000000000000002e163 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. unpow2 67.5%

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

   4. Simplified 67.5%

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

      1. times-frac 80.5%

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

      2. div-inv 80.5%

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

      3. metadata-eval 80.5%

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

   6. Applied egg-rr 80.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.4e+163) (* 0.5 (- y (* z (/ z y)))) (* (/ x y) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.4e+163) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.4d+163) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.4e+163) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.4e+163:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.4e+163)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.4e+163)
		tmp = 0.5 * (y - (z * (z / y)));
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.4e+163], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.39999999999999973e163

   1. Initial program 69.7%

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

      1. div-sub 63.8%

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

      2. sub-neg 63.8%

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

      3. add-sqr-sqrt 63.8%

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

      4. times-frac 63.9%

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

      5. fma-def 65.6%

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

      6. hypot-def 65.6%

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

      7. hypot-def 86.3%

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

      8. times-frac 94.4%

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

   3. Applied egg-rr 94.4%

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

   4. Taylor expanded in x around 0 68.8%

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

      1. distribute-lft-out-- 68.8%

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

      2. unpow2 68.8%

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

      3. associate-*r/ 75.6%

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

   6. Simplified 75.6%

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

   if 4.39999999999999973e163 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. unpow2 67.5%

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

   4. Simplified 67.5%

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

      1. times-frac 80.5%

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

      2. div-inv 80.5%

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

      3. metadata-eval 80.5%

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

   6. Applied egg-rr 80.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e+163) (* 0.5 (- y (/ z (/ y z)))) (* (/ x y) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e+163) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4d+163) then
        tmp = 0.5d0 * (y - (z / (y / z)))
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e+163) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4e+163:
		tmp = 0.5 * (y - (z / (y / z)))
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e+163)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4e+163)
		tmp = 0.5 * (y - (z / (y / z)));
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e+163], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e163

   1. Initial program 69.7%

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

   2. Taylor expanded in x around inf 79.6%

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

      1. +-commutative 79.6%

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

      2. distribute-lft-out 79.6%

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

      3. unpow2 79.6%

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

      4. associate-/l* 83.8%

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

      5. unpow2 83.8%

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

      6. associate-/l* 91.8%

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

   4. Simplified 91.8%

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

   5. Taylor expanded in x around 0 68.8%

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

      1. unpow2 68.8%

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

      2. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

   if 3.9999999999999998e163 < x

   1. Initial program 52.1%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. unpow2 67.5%

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

   4. Simplified 67.5%

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

      1. times-frac 80.5%

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

      2. div-inv 80.5%

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

      3. metadata-eval 80.5%

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

   6. Applied egg-rr 80.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+163}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+68) (* y 0.5) (* 0.5 (/ x (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+68) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5d+68) then
        tmp = y * 0.5d0
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+68) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5e+68:
		tmp = y * 0.5
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+68)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5e+68)
		tmp = y * 0.5;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5e+68], 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 < 5.0000000000000004e68

   1. Initial program 69.7%

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

   2. Taylor expanded in y around inf 43.7%

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

   if 5.0000000000000004e68 < 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 49.2%

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

      1. unpow2 49.2%

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

      2. associate-/l* 54.0%

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

   4. Simplified 54.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 14: 34.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

2. Taylor expanded in y around inf 40.9%

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

3. Final simplification 40.9%

   \[\leadsto y \cdot 0.5 \]

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 2023293 
(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)))