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

Percentage Accurate: 68.5% → 95.4%

Time: 7.3s

Alternatives: 10

Speedup: 1.1×

• 42.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+297}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{1}{y \cdot \frac{2}{x + z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+151)
   (* 0.5 (+ (/ x (/ y x)) (- y (/ z (/ y z)))))
   (if (<= (* x x) 2e+297)
     (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))
     (* (- x z) (/ 1.0 (* y (/ 2.0 (+ x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+151) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else if ((x * x) <= 2e+297) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = (x - z) * (1.0 / (y * (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) <= 2d+151) then
        tmp = 0.5d0 * ((x / (y / x)) + (y - (z / (y / z))))
    else if ((x * x) <= 2d+297) then
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    else
        tmp = (x - z) * (1.0d0 / (y * (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) <= 2e+151) {
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	} else if ((x * x) <= 2e+297) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+151:
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))))
	elif (x * x) <= 2e+297:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	else:
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+151)
		tmp = Float64(0.5 * Float64(Float64(x / Float64(y / x)) + Float64(y - Float64(z / Float64(y / z)))));
	elseif (Float64(x * x) <= 2e+297)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	else
		tmp = Float64(Float64(x - z) * Float64(1.0 / Float64(y * Float64(2.0 / Float64(x + z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+151)
		tmp = 0.5 * ((x / (y / x)) + (y - (z / (y / z))));
	elseif ((x * x) <= 2e+297)
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	else
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+151], N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+297], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(1.0 / N[(y * N[(2.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.00000000000000003e151

   1. Initial program 79.5%

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

   2. Taylor expanded in x around inf 94.5%

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

      1. +-commutative 94.5%

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

      2. distribute-lft-out 94.5%

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

      3. unpow2 94.5%

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

      4. associate-/l* 95.6%

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

      5. unpow2 95.6%

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

      6. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   if 2.00000000000000003e151 < (*.f64 x x) < 2e297

   1. Initial program 68.2%

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

   2. Taylor expanded in y around 0 96.7%

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

      1. distribute-lft-out 96.7%

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

      2. unpow2 96.7%

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

      3. unpow2 96.7%

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

   4. Simplified 96.7%

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

   if 2e297 < (*.f64 x x)

   1. Initial program 66.2%

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

   2. Taylor expanded in y around 0 66.2%

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

      1. unpow2 66.2%

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

      2. unpow2 66.2%

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

   4. Simplified 66.2%

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

      1. difference-of-squares 85.2%

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

      2. *-commutative 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. associate-/l* 96.0%

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

      2. div-inv 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

      5. /-rgt-identity 95.9%

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

   8. Applied egg-rr 95.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+297}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{1}{y \cdot \frac{2}{x + z}}\\ \end{array} \]

Alternative 2: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e-180)
   (* 0.5 (/ (- y z) (/ y (+ y z))))
   (if (<= (* x x) 2e+305)
     (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))
     (* 0.5 (+ y (* x (/ x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-180) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 2e+305) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} 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) <= 5d-180) then
        tmp = 0.5d0 * ((y - z) / (y / (y + z)))
    else if ((x * x) <= 2d+305) then
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    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) <= 5e-180) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 2e+305) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e-180:
		tmp = 0.5 * ((y - z) / (y / (y + z)))
	elif (x * x) <= 2e+305:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	else:
		tmp = 0.5 * (y + (x * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e-180)
		tmp = Float64(0.5 * Float64(Float64(y - z) / Float64(y / Float64(y + z))));
	elseif (Float64(x * x) <= 2e+305)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e-180)
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	elseif ((x * x) <= 2e+305)
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	else
		tmp = 0.5 * (y + (x * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-180], N[(0.5 * N[(N[(y - z), $MachinePrecision] / N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+305], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000001e-180

   1. Initial program 76.6%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. unpow2 75.7%

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

      2. unpow2 75.7%

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

      3. difference-of-squares 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-/l* 96.9%

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

   4. Simplified 96.9%

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

   if 5.0000000000000001e-180 < (*.f64 x x) < 1.9999999999999999e305

   1. Initial program 79.1%

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

   2. Taylor expanded in y around 0 98.0%

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

      1. distribute-lft-out 98.0%

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

      2. unpow2 98.0%

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

      3. unpow2 98.0%

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

   4. Simplified 98.0%

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

   if 1.9999999999999999e305 < (*.f64 x x)

   1. Initial program 65.2%

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

   2. Taylor expanded in y around 0 65.2%

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

      1. distribute-lft-out 65.2%

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

      2. unpow2 65.2%

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

      3. unpow2 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in x around inf 80.3%

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

      1. unpow2 80.3%

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

      2. associate-*r/ 95.5%

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

   7. Simplified 95.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 3: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+297}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{1}{y \cdot \frac{2}{x + z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e-180)
   (* 0.5 (/ (- y z) (/ y (+ y z))))
   (if (<= (* x x) 2e+297)
     (* 0.5 (+ y (/ (- (* x x) (* z z)) y)))
     (* (- x z) (/ 1.0 (* y (/ 2.0 (+ x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-180) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 2e+297) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = (x - z) * (1.0 / (y * (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-180) then
        tmp = 0.5d0 * ((y - z) / (y / (y + z)))
    else if ((x * x) <= 2d+297) then
        tmp = 0.5d0 * (y + (((x * x) - (z * z)) / y))
    else
        tmp = (x - z) * (1.0d0 / (y * (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-180) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 2e+297) {
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	} else {
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e-180:
		tmp = 0.5 * ((y - z) / (y / (y + z)))
	elif (x * x) <= 2e+297:
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y))
	else:
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e-180)
		tmp = Float64(0.5 * Float64(Float64(y - z) / Float64(y / Float64(y + z))));
	elseif (Float64(x * x) <= 2e+297)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(Float64(x * x) - Float64(z * z)) / y)));
	else
		tmp = Float64(Float64(x - z) * Float64(1.0 / Float64(y * 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-180)
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	elseif ((x * x) <= 2e+297)
		tmp = 0.5 * (y + (((x * x) - (z * z)) / y));
	else
		tmp = (x - z) * (1.0 / (y * (2.0 / (x + z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-180], N[(0.5 * N[(N[(y - z), $MachinePrecision] / N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+297], N[(0.5 * N[(y + N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(1.0 / N[(y * N[(2.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000001e-180

   1. Initial program 76.6%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. unpow2 75.7%

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

      2. unpow2 75.7%

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

      3. difference-of-squares 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-/l* 96.9%

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

   4. Simplified 96.9%

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

   if 5.0000000000000001e-180 < (*.f64 x x) < 2e297

   1. Initial program 78.7%

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

   2. Taylor expanded in y around 0 97.9%

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

      1. distribute-lft-out 97.9%

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

      2. unpow2 97.9%

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

      3. unpow2 97.9%

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

   4. Simplified 97.9%

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

   if 2e297 < (*.f64 x x)

   1. Initial program 66.2%

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

   2. Taylor expanded in y around 0 66.2%

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

      1. unpow2 66.2%

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

      2. unpow2 66.2%

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

   4. Simplified 66.2%

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

      1. difference-of-squares 85.2%

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

      2. *-commutative 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. associate-/l* 96.0%

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

      2. div-inv 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

      5. /-rgt-identity 95.9%

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

   8. Applied egg-rr 95.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+297}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x - z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{1}{y \cdot \frac{2}{x + z}}\\ \end{array} \]

Alternative 4: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \left(x + z\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4e-111)
   (* 0.5 (/ (- y z) (/ y (+ y z))))
   (if (<= (* x x) 5e+263)
     (/ (* (- x z) (+ x z)) (* y 2.0))
     (* 0.5 (+ y (* x (/ x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e-111) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 5e+263) {
		tmp = ((x - z) * (x + z)) / (y * 2.0);
	} 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) <= 4d-111) then
        tmp = 0.5d0 * ((y - z) / (y / (y + z)))
    else if ((x * x) <= 5d+263) then
        tmp = ((x - z) * (x + z)) / (y * 2.0d0)
    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) <= 4e-111) {
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	} else if ((x * x) <= 5e+263) {
		tmp = ((x - z) * (x + z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 4e-111:
		tmp = 0.5 * ((y - z) / (y / (y + z)))
	elif (x * x) <= 5e+263:
		tmp = ((x - z) * (x + z)) / (y * 2.0)
	else:
		tmp = 0.5 * (y + (x * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4e-111)
		tmp = Float64(0.5 * Float64(Float64(y - z) / Float64(y / Float64(y + z))));
	elseif (Float64(x * x) <= 5e+263)
		tmp = Float64(Float64(Float64(x - z) * Float64(x + z)) / Float64(y * 2.0));
	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) <= 4e-111)
		tmp = 0.5 * ((y - z) / (y / (y + z)));
	elseif ((x * x) <= 5e+263)
		tmp = ((x - z) * (x + z)) / (y * 2.0);
	else
		tmp = 0.5 * (y + (x * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-111], N[(0.5 * N[(N[(y - z), $MachinePrecision] / N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+263], N[(N[(N[(x - z), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.00000000000000035e-111

   1. Initial program 77.5%

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

   2. Taylor expanded in x around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

      3. difference-of-squares 76.9%

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

      4. *-commutative 76.9%

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

      5. associate-/l* 97.3%

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

   4. Simplified 97.3%

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

   if 4.00000000000000035e-111 < (*.f64 x x) < 5.00000000000000022e263

   1. Initial program 81.5%

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

   2. Taylor expanded in y around 0 78.0%

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

      1. unpow2 78.0%

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

      2. unpow2 78.0%

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

   4. Simplified 78.0%

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

      1. difference-of-squares 78.0%

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

      2. *-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

   if 5.00000000000000022e263 < (*.f64 x x)

   1. Initial program 64.5%

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

   2. Taylor expanded in y around 0 70.5%

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

      1. distribute-lft-out 70.5%

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

      2. unpow2 70.5%

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

      3. unpow2 70.5%

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

   4. Simplified 70.5%

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

   5. Taylor expanded in x around inf 80.7%

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

      1. unpow2 80.7%

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

      2. associate-*r/ 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-111}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{\frac{y}{y + z}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \left(x + z\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+258}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+258) (* 0.5 (+ y (* x (/ x y)))) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+258) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = z * ((z / 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 ((z * z) <= 5d+258) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = z * ((z / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+258) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+258:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+258)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+258)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+258], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e258

   1. Initial program 77.7%

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

   2. Taylor expanded in y around 0 94.0%

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

      1. distribute-lft-out 94.0%

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

      2. unpow2 94.0%

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

      3. unpow2 94.0%

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

   4. Simplified 94.0%

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

   5. Taylor expanded in x around inf 77.9%

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

      1. unpow2 77.9%

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

      2. associate-*r/ 83.8%

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

   7. Simplified 83.8%

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

   if 5e258 < (*.f64 z z)

   1. Initial program 68.2%

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

   2. Taylor expanded in z around inf 71.8%

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

      1. unpow2 71.8%

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

      2. associate-*r/ 71.8%

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

   4. Simplified 71.8%

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

   5. Taylor expanded in z around 0 71.8%

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

      1. unpow2 71.8%

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

      2. associate-*r/ 75.8%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. associate-*l* 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+258}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]

Alternative 6: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+169)
   (* 0.5 (- y (* z (/ z y))))
   (* 0.5 (+ y (* x (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+169) {
		tmp = 0.5 * (y - (z * (z / y)));
	} 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+169) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    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+169) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+169:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = 0.5 * (y + (x * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+169)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	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+169)
		tmp = 0.5 * (y - (z * (z / y)));
	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+169], N[(0.5 * N[(y - N[(z * N[(z / y), $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) < 1.99999999999999987e169

   1. Initial program 79.4%

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

   2. Taylor expanded in x around inf 94.0%

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

      1. +-commutative 94.0%

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

      2. distribute-lft-out 94.0%

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

      3. unpow2 94.0%

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

      4. associate-/l* 95.1%

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

      5. unpow2 95.1%

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

      6. associate-/l* 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 84.0%

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

      1. unpow2 84.0%

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

      2. associate-*r/ 88.2%

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

   7. Simplified 88.2%

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

   if 1.99999999999999987e169 < (*.f64 x x)

   1. Initial program 66.5%

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

   2. Taylor expanded in y around 0 74.4%

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

      1. distribute-lft-out 74.4%

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

      2. unpow2 74.4%

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

      3. unpow2 74.4%

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

   4. Simplified 74.4%

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

   5. Taylor expanded in x around inf 79.8%

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

      1. unpow2 79.8%

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

      2. associate-*r/ 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 7: 52.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+169) (* z (* (/ z y) -0.5)) (* 0.5 (* x (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+169) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = 0.5 * (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+169) then
        tmp = z * ((z / y) * (-0.5d0))
    else
        tmp = 0.5d0 * (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+169) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+169:
		tmp = z * ((z / y) * -0.5)
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+169)
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+169)
		tmp = z * ((z / y) * -0.5);
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+169], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999987e169

   1. Initial program 79.4%

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

   2. Taylor expanded in z around inf 49.7%

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

      1. unpow2 49.7%

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

      2. associate-*r/ 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in z around 0 49.7%

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

      1. unpow2 49.7%

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

      2. associate-*r/ 51.2%

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

      3. associate-*r* 51.2%

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

      4. *-commutative 51.2%

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

      5. associate-*l* 51.2%

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

   7. Simplified 51.2%

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

   if 1.99999999999999987e169 < (*.f64 x x)

   1. Initial program 66.5%

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

   2. Taylor expanded in x around inf 65.0%

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

      1. +-commutative 65.0%

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

      2. distribute-lft-out 65.0%

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

      3. unpow2 65.0%

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

      4. associate-/l* 74.8%

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

      5. unpow2 74.8%

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

      6. associate-/l* 77.8%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in x around inf 72.3%

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

      1. unpow2 72.3%

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

      2. associate-*r/ 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 8: 52.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+169) (/ (* z -0.5) (/ y z)) (* 0.5 (* x (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+169) {
		tmp = (z * -0.5) / (y / z);
	} else {
		tmp = 0.5 * (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+169) then
        tmp = (z * (-0.5d0)) / (y / z)
    else
        tmp = 0.5d0 * (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+169) {
		tmp = (z * -0.5) / (y / z);
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+169:
		tmp = (z * -0.5) / (y / z)
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+169)
		tmp = Float64(Float64(z * -0.5) / Float64(y / z));
	else
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+169)
		tmp = (z * -0.5) / (y / z);
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+169], N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999987e169

   1. Initial program 79.4%

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

   2. Taylor expanded in z around inf 49.7%

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

      1. unpow2 49.7%

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

      2. associate-*r/ 49.7%

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

   4. Simplified 49.7%

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

   5. Taylor expanded in z around 0 49.7%

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

      1. unpow2 49.7%

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

      2. associate-*r/ 51.2%

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

      3. associate-*r* 51.2%

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

      4. *-commutative 51.2%

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

      5. associate-*l* 51.2%

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

   7. Simplified 51.2%

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

      1. associate-*r* 51.2%

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

      2. clear-num 51.2%

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

      3. un-div-inv 51.2%

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

   9. Applied egg-rr 51.2%

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

   if 1.99999999999999987e169 < (*.f64 x x)

   1. Initial program 66.5%

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

   2. Taylor expanded in x around inf 65.0%

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

      1. +-commutative 65.0%

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

      2. distribute-lft-out 65.0%

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

      3. unpow2 65.0%

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

      4. associate-/l* 74.8%

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

      5. unpow2 74.8%

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

      6. associate-/l* 77.8%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in x around inf 72.3%

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

      1. unpow2 72.3%

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

      2. associate-*r/ 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 9: 43.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.85e-53) (* 0.5 y) (* 0.5 (* x (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.85e-53) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (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 <= 2.85d-53) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (x * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.85e-53) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.85e-53:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.85e-53)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.85e-53)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.85e-53], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8500000000000001e-53

   1. Initial program 73.3%

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

   2. Taylor expanded in y around inf 33.9%

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

   if 2.8500000000000001e-53 < x

   1. Initial program 77.7%

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

   2. Taylor expanded in x around inf 81.3%

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

      1. +-commutative 81.3%

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

      2. distribute-lft-out 81.3%

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

      3. unpow2 81.3%

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

      4. associate-/l* 86.4%

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

      5. unpow2 86.4%

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

      6. associate-/l* 89.0%

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

   4. Simplified 89.0%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. unpow2 62.0%

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

      2. associate-*r/ 65.8%

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

   7. Simplified 65.8%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 10: 34.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

2. Taylor expanded in y around inf 28.5%

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

3. Final simplification 28.5%

   \[\leadsto 0.5 \cdot y \]

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 2023268 
(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)))