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

Percentage Accurate: 69.0% → 90.5%

Time: 11.3s

Alternatives: 8

Speedup: 1.4×

• 41.4% 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: 69.0% 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: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.2e+161)
   (* 0.5 (- y (/ z (/ y z))))
   (if (<= y 3.2e+150)
     (/ (fma x x (* (+ y z) (- y z))) (* y 2.0))
     (* 0.5 (* (- y z) (/ (+ y z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.2e+161) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else if (y <= 3.2e+150) {
		tmp = fma(x, x, ((y + z) * (y - z))) / (y * 2.0);
	} else {
		tmp = 0.5 * ((y - z) * ((y + z) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.2e+161)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	elseif (y <= 3.2e+150)
		tmp = Float64(fma(x, x, Float64(Float64(y + z) * Float64(y - z))) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(y - z) * Float64(Float64(y + z) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.2e+161], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+150], N[(N[(x * x + N[(N[(y + z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y - z), $MachinePrecision] * N[(N[(y + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.20000000000000013e161

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0 8.8%

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

      1. unpow2 8.8%

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

      2. unpow2 8.8%

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

      3. div-sub 8.8%

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

      4. associate-/l* 77.4%

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

      5. *-inverses 77.4%

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

      6. /-rgt-identity 77.4%

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

      7. associate-/l* 96.4%

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

   4. Simplified 96.4%

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

   if -6.20000000000000013e161 < y < 3.20000000000000016e150

   1. Initial program 91.1%

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

      1. associate--l+ 91.1%

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

      2. fma-def 93.6%

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

      3. difference-of-squares 93.6%

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

   3. Simplified 93.6%

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

   if 3.20000000000000016e150 < y

   1. Initial program 11.7%

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

   2. Taylor expanded in x around 0 11.7%

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

      1. unpow2 11.7%

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

      2. unpow2 11.7%

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

      3. difference-of-squares 16.3%

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

      4. associate-/l* 96.8%

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

      5. associate-/r/ 96.9%

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

   4. Simplified 96.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z\right) \cdot \frac{y + z}{y}\right)\\ \end{array} \]

Alternative 2: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+161} \lor \neg \left(y \leq 3.2 \cdot 10^{+99}\right):\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.2e+161) (not (<= y 3.2e+99)))
   (* 0.5 (- 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 <= -6.2e+161) || !(y <= 3.2e+99)) {
		tmp = 0.5 * (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 <= (-6.2d+161)) .or. (.not. (y <= 3.2d+99))) then
        tmp = 0.5d0 * (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 <= -6.2e+161) || !(y <= 3.2e+99)) {
		tmp = 0.5 * (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 <= -6.2e+161) or not (y <= 3.2e+99):
		tmp = 0.5 * (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 <= -6.2e+161) || !(y <= 3.2e+99))
		tmp = Float64(0.5 * 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 <= -6.2e+161) || ~((y <= 3.2e+99)))
		tmp = 0.5 * (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, -6.2e+161], N[Not[LessEqual[y, 3.2e+99]], $MachinePrecision]], N[(0.5 * N[(y - N[(z / N[(y / z), $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 < -6.20000000000000013e161 or 3.19999999999999999e99 < y

   1. Initial program 20.8%

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

   2. Taylor expanded in x around 0 18.1%

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

      1. unpow2 18.1%

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

      2. unpow2 18.1%

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

      3. div-sub 18.1%

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

      4. associate-/l* 75.4%

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

      5. *-inverses 75.4%

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

      6. /-rgt-identity 75.4%

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

      7. associate-/l* 93.0%

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

   4. Simplified 93.0%

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

   if -6.20000000000000013e161 < y < 3.19999999999999999e99

   1. Initial program 92.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+161} \lor \neg \left(y \leq 3.2 \cdot 10^{+99}\right):\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 3: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ t_1 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))) (t_1 (* -0.5 (* z (/ z y)))))
   (if (<= y -2.35e+14)
     (* y 0.5)
     (if (<= y -1.8e-48)
       t_1
       (if (<= y -6e-179)
         t_0
         (if (<= y -1.65e-206) t_1 (if (<= y 1.25e+28) t_0 (* y 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double t_1 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -2.35e+14) {
		tmp = y * 0.5;
	} else if (y <= -1.8e-48) {
		tmp = t_1;
	} else if (y <= -6e-179) {
		tmp = t_0;
	} else if (y <= -1.65e-206) {
		tmp = t_1;
	} else if (y <= 1.25e+28) {
		tmp = t_0;
	} else {
		tmp = y * 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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (x / (y / x))
    t_1 = (-0.5d0) * (z * (z / y))
    if (y <= (-2.35d+14)) then
        tmp = y * 0.5d0
    else if (y <= (-1.8d-48)) then
        tmp = t_1
    else if (y <= (-6d-179)) then
        tmp = t_0
    else if (y <= (-1.65d-206)) then
        tmp = t_1
    else if (y <= 1.25d+28) then
        tmp = t_0
    else
        tmp = y * 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 t_1 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -2.35e+14) {
		tmp = y * 0.5;
	} else if (y <= -1.8e-48) {
		tmp = t_1;
	} else if (y <= -6e-179) {
		tmp = t_0;
	} else if (y <= -1.65e-206) {
		tmp = t_1;
	} else if (y <= 1.25e+28) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	t_1 = -0.5 * (z * (z / y))
	tmp = 0
	if y <= -2.35e+14:
		tmp = y * 0.5
	elif y <= -1.8e-48:
		tmp = t_1
	elif y <= -6e-179:
		tmp = t_0
	elif y <= -1.65e-206:
		tmp = t_1
	elif y <= 1.25e+28:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	t_1 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (y <= -2.35e+14)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.8e-48)
		tmp = t_1;
	elseif (y <= -6e-179)
		tmp = t_0;
	elseif (y <= -1.65e-206)
		tmp = t_1;
	elseif (y <= 1.25e+28)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x / (y / x));
	t_1 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (y <= -2.35e+14)
		tmp = y * 0.5;
	elseif (y <= -1.8e-48)
		tmp = t_1;
	elseif (y <= -6e-179)
		tmp = t_0;
	elseif (y <= -1.65e-206)
		tmp = t_1;
	elseif (y <= 1.25e+28)
		tmp = t_0;
	else
		tmp = y * 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]}, Block[{t$95$1 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.35e+14], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.8e-48], t$95$1, If[LessEqual[y, -6e-179], t$95$0, If[LessEqual[y, -1.65e-206], t$95$1, If[LessEqual[y, 1.25e+28], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.35e14 or 1.24999999999999989e28 < y

   1. Initial program 52.2%

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

   2. Taylor expanded in y around inf 75.3%

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

   if -2.35e14 < y < -1.8000000000000001e-48 or -6.00000000000000012e-179 < y < -1.6499999999999999e-206

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 74.3%

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

      1. *-commutative 74.3%

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

      2. unpow2 74.3%

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

      3. associate-*l/ 74.2%

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

      4. *-commutative 74.2%

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

   4. Simplified 74.2%

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

   if -1.8000000000000001e-48 < y < -6.00000000000000012e-179 or -1.6499999999999999e-206 < y < 1.24999999999999989e28

   1. Initial program 89.5%

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

   2. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

      2. associate-/l* 57.1%

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

   4. Simplified 57.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-179}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-206}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{+14}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-206}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))))
   (if (<= y -1.26e+14)
     (* y 0.5)
     (if (<= y -1.45e-48)
       (* -0.5 (* z (/ z y)))
       (if (<= y -1.2e-181)
         t_0
         (if (<= y -2e-206)
           (* -0.5 (/ (* z z) y))
           (if (<= y 4.8e+26) t_0 (* y 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (y <= -1.26e+14) {
		tmp = y * 0.5;
	} else if (y <= -1.45e-48) {
		tmp = -0.5 * (z * (z / y));
	} else if (y <= -1.2e-181) {
		tmp = t_0;
	} else if (y <= -2e-206) {
		tmp = -0.5 * ((z * z) / y);
	} else if (y <= 4.8e+26) {
		tmp = t_0;
	} else {
		tmp = y * 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 (y <= (-1.26d+14)) then
        tmp = y * 0.5d0
    else if (y <= (-1.45d-48)) then
        tmp = (-0.5d0) * (z * (z / y))
    else if (y <= (-1.2d-181)) then
        tmp = t_0
    else if (y <= (-2d-206)) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (y <= 4.8d+26) then
        tmp = t_0
    else
        tmp = y * 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 (y <= -1.26e+14) {
		tmp = y * 0.5;
	} else if (y <= -1.45e-48) {
		tmp = -0.5 * (z * (z / y));
	} else if (y <= -1.2e-181) {
		tmp = t_0;
	} else if (y <= -2e-206) {
		tmp = -0.5 * ((z * z) / y);
	} else if (y <= 4.8e+26) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	tmp = 0
	if y <= -1.26e+14:
		tmp = y * 0.5
	elif y <= -1.45e-48:
		tmp = -0.5 * (z * (z / y))
	elif y <= -1.2e-181:
		tmp = t_0
	elif y <= -2e-206:
		tmp = -0.5 * ((z * z) / y)
	elif y <= 4.8e+26:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (y <= -1.26e+14)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.45e-48)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	elseif (y <= -1.2e-181)
		tmp = t_0;
	elseif (y <= -2e-206)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (y <= 4.8e+26)
		tmp = t_0;
	else
		tmp = Float64(y * 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 (y <= -1.26e+14)
		tmp = y * 0.5;
	elseif (y <= -1.45e-48)
		tmp = -0.5 * (z * (z / y));
	elseif (y <= -1.2e-181)
		tmp = t_0;
	elseif (y <= -2e-206)
		tmp = -0.5 * ((z * z) / y);
	elseif (y <= 4.8e+26)
		tmp = t_0;
	else
		tmp = y * 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[y, -1.26e+14], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.45e-48], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.2e-181], t$95$0, If[LessEqual[y, -2e-206], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+26], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.26e14 or 4.80000000000000009e26 < y

   1. Initial program 52.2%

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

   2. Taylor expanded in y around inf 75.3%

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

   if -1.26e14 < y < -1.4500000000000001e-48

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 73.3%

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

      1. *-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. associate-*l/ 73.3%

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

      4. *-commutative 73.3%

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

   4. Simplified 73.3%

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

   if -1.4500000000000001e-48 < y < -1.2000000000000001e-181 or -2.00000000000000006e-206 < y < 4.80000000000000009e26

   1. Initial program 89.5%

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

   2. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

      2. associate-/l* 57.1%

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

   4. Simplified 57.1%

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

   if -1.2000000000000001e-181 < y < -2.00000000000000006e-206

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf 75.7%

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

      1. *-commutative 75.7%

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

      2. unpow2 75.7%

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

   4. Simplified 75.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+14}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-206}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+113)
   (* 0.5 (- y (/ z (/ y z))))
   (* 0.5 (+ y (* x (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+113) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+113) then
        tmp = 0.5d0 * (y - (z / (y / z)))
    else
        tmp = 0.5d0 * (y + (x * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+113) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+113:
		tmp = 0.5 * (y - (z / (y / z)))
	else:
		tmp = 0.5 * (y + (x * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+113)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+113)
		tmp = 0.5 * (y - (z / (y / z)));
	else
		tmp = 0.5 * (y + (x * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+113], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e113

   1. Initial program 75.7%

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

   2. Taylor expanded in x around 0 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. div-sub 66.8%

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

      4. associate-/l* 84.0%

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

      5. *-inverses 84.0%

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

      6. /-rgt-identity 84.0%

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

      7. associate-/l* 90.4%

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

   4. Simplified 90.4%

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

   if 2e113 < (*.f64 x x)

   1. Initial program 70.1%

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

   2. Taylor expanded in z around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

   4. Simplified 71.1%

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

   5. Taylor expanded in x around 0 79.3%

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

      1. distribute-lft-out 79.3%

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

      2. unpow2 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*l/ 86.4%

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

   9. Applied egg-rr 86.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 52.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.6e-16)
   (* y 0.5)
   (if (<= y 2.2e+27) (* 0.5 (/ x (/ y x))) (* y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e-16) {
		tmp = y * 0.5;
	} else if (y <= 2.2e+27) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = y * 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 (y <= (-7.6d-16)) then
        tmp = y * 0.5d0
    else if (y <= 2.2d+27) then
        tmp = 0.5d0 * (x / (y / x))
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e-16) {
		tmp = y * 0.5;
	} else if (y <= 2.2e+27) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.6e-16:
		tmp = y * 0.5
	elif y <= 2.2e+27:
		tmp = 0.5 * (x / (y / x))
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.6e-16)
		tmp = Float64(y * 0.5);
	elseif (y <= 2.2e+27)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.6e-16)
		tmp = y * 0.5;
	elseif (y <= 2.2e+27)
		tmp = 0.5 * (x / (y / x));
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.6e-16], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 2.2e+27], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.60000000000000024e-16 or 2.1999999999999999e27 < y

   1. Initial program 55.0%

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

   2. Taylor expanded in y around inf 72.6%

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

   if -7.60000000000000024e-16 < y < 2.1999999999999999e27

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 52.9%

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

      1. unpow2 52.9%

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

      2. associate-/l* 53.6%

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

   4. Simplified 53.6%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 8.2e+169) (* 0.5 (+ y (* x (/ x y)))) (* (/ z (/ y z)) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8.2e+169) {
		tmp = 0.5 * (y + (x * (x / y)));
	} 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) :: tmp
    if (z <= 8.2d+169) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    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 tmp;
	if (z <= 8.2e+169) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 8.2e+169:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = (z / (y / z)) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8.2e+169)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(Float64(z / Float64(y / z)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 8.2e+169)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = (z / (y / z)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8.2e+169], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.2000000000000006e169

   1. Initial program 74.7%

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

   2. Taylor expanded in z around 0 56.7%

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

      1. unpow2 56.7%

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

      2. unpow2 56.7%

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

   4. Simplified 56.7%

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

   5. Taylor expanded in x around 0 73.1%

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

      1. distribute-lft-out 73.1%

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

      2. unpow2 73.1%

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

   7. Simplified 73.1%

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

      1. associate-*l/ 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 8.2000000000000006e169 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. associate-/l* 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \]

Alternative 8: 34.9% 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 73.4%

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

2. Taylor expanded in y around inf 38.6%

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

3. Final simplification 38.6%

   \[\leadsto y \cdot 0.5 \]

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 2023275 
(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)))