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

Percentage Accurate: 70.1% → 99.9%

Time: 7.5s

Alternatives: 8

Speedup: 1.4×

• 42.4% 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: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ y (/ (+ z x) (/ y (- x z))))))

C

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

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 + ((z + x) / (y / (x - z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

2. Taylor expanded in y around 0 80.3%

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

   1. +-lft-identity 80.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   2. +-commutative 80.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   3. mul0-lft 80.3%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   4. metadata-eval 80.3%

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

   5. distribute-rgt1-in 80.3%

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

   6. distribute-lft-out 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. metadata-eval 80.3%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   9. mul0-lft 80.3%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   10. +-rgt-identity 80.3%

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

   11. unpow2 80.3%

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

   12. unpow2 80.3%

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

   13. difference-of-squares 86.6%

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

   14. associate-/l* 99.9%

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

   15. +-commutative 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 53.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ -0.5 y)))))
   (if (<= (* x x) 5e-154)
     t_0
     (if (<= (* x x) 5e-79)
       (* 0.5 y)
       (if (<= (* x x) 0.02)
         t_0
         (if (<= (* x x) 1e+155) (* 0.5 y) (* 0.5 (* x (/ x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = t_0;
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = t_0;
	} else if ((x * x) <= 1e+155) {
		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) :: t_0
    real(8) :: tmp
    t_0 = z * (z * ((-0.5d0) / y))
    if ((x * x) <= 5d-154) then
        tmp = t_0
    else if ((x * x) <= 5d-79) then
        tmp = 0.5d0 * y
    else if ((x * x) <= 0.02d0) then
        tmp = t_0
    else if ((x * x) <= 1d+155) 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 t_0 = z * (z * (-0.5 / y));
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = t_0;
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = t_0;
	} else if ((x * x) <= 1e+155) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * (-0.5 / y))
	tmp = 0
	if (x * x) <= 5e-154:
		tmp = t_0
	elif (x * x) <= 5e-79:
		tmp = 0.5 * y
	elif (x * x) <= 0.02:
		tmp = t_0
	elif (x * x) <= 1e+155:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * Float64(-0.5 / y)))
	tmp = 0.0
	if (Float64(x * x) <= 5e-154)
		tmp = t_0;
	elseif (Float64(x * x) <= 5e-79)
		tmp = Float64(0.5 * y);
	elseif (Float64(x * x) <= 0.02)
		tmp = t_0;
	elseif (Float64(x * x) <= 1e+155)
		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)
	t_0 = z * (z * (-0.5 / y));
	tmp = 0.0;
	if ((x * x) <= 5e-154)
		tmp = t_0;
	elseif ((x * x) <= 5e-79)
		tmp = 0.5 * y;
	elseif ((x * x) <= 0.02)
		tmp = t_0;
	elseif ((x * x) <= 1e+155)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-154], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 5e-79], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 0.02], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+155], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000002e-154 or 4.99999999999999999e-79 < (*.f64 x x) < 0.0200000000000000004

   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 49.3%

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

      1. *-commutative 49.3%

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

      2. unpow2 49.3%

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

      3. associate-/l* 56.1%

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

   4. Simplified 56.1%

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

   5. Taylor expanded in z around 0 49.3%

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

      1. unpow2 49.3%

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

      2. associate-/l* 56.1%

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

      3. associate-*r/ 56.1%

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

      4. associate-*l/ 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-/r/ 56.1%

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

   7. Simplified 56.1%

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

   if 5.0000000000000002e-154 < (*.f64 x x) < 4.99999999999999999e-79 or 0.0200000000000000004 < (*.f64 x x) < 1.00000000000000001e155

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 55.1%

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

   if 1.00000000000000001e155 < (*.f64 x x)

   1. Initial program 65.3%

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

   2. Taylor expanded in x around inf 64.3%

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

      1. unpow2 64.3%

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

      2. associate-/l* 69.0%

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

   4. Simplified 69.0%

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

      1. associate-/r/ 69.0%

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

   6. Applied egg-rr 69.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{z + x}{\frac{y}{x - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-9)
   (* 0.5 (+ y (/ x (/ y x))))
   (* 0.5 (/ (+ z x) (/ y (- x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * ((z + x) / (y / (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 ((z * z) <= 5d-9) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = 0.5d0 * ((z + x) / (y / (x - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * ((z + x) / (y / (x - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-9:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = 0.5 * ((z + x) / (y / (x - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-9)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(0.5 * Float64(Float64(z + x) / Float64(y / Float64(x - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-9)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = 0.5 * ((z + x) / (y / (x - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-9], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(z + x), $MachinePrecision] / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-9

   1. Initial program 73.8%

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

   2. Taylor expanded in y around 0 89.7%

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

      1. +-lft-identity 89.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 89.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 89.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 89.7%

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

      5. distribute-rgt1-in 89.7%

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

      6. distribute-lft-out 89.7%

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

      7. distribute-rgt1-in 89.7%

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

      8. metadata-eval 89.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 89.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 89.7%

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

      11. unpow2 89.7%

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

      12. unpow2 89.7%

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

      13. difference-of-squares 89.7%

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

      14. associate-/l* 99.9%

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

      15. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 82.7%

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

      1. unpow2 82.7%

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

      2. associate-/l* 91.5%

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

   7. Simplified 91.5%

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

   if 5.0000000000000001e-9 < (*.f64 z z)

   1. Initial program 61.2%

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

   2. Taylor expanded in y around 0 64.1%

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

      1. unpow2 64.1%

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

      2. unpow2 64.1%

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

      3. sub-neg 64.1%

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

      4. mul-1-neg 64.1%

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

      5. unpow2 64.1%

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

      6. +-commutative 64.1%

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

      7. unpow2 64.1%

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

      8. unpow2 64.1%

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

      9. +-commutative 64.1%

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

      10. unpow2 64.1%

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

      11. mul-1-neg 64.1%

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

      12. sub-neg 64.1%

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

      13. difference-of-squares 76.6%

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

      14. associate-/l* 83.8%

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

      15. +-commutative 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{z + x}{\frac{y}{x - z}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ y (* (- x z) (/ (+ z x) y)))))

C

double code(double x, double y, double z) {
	return 0.5 * (y + ((x - z) * ((z + x) / 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 + ((x - z) * ((z + x) / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

2. Taylor expanded in y around 0 80.3%

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

   1. +-lft-identity 80.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   2. +-commutative 80.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   3. mul0-lft 80.3%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   4. metadata-eval 80.3%

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

   5. distribute-rgt1-in 80.3%

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

   6. distribute-lft-out 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. metadata-eval 80.3%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   9. mul0-lft 80.3%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   10. +-rgt-identity 80.3%

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

   11. unpow2 80.3%

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

   12. unpow2 80.3%

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

   13. difference-of-squares 86.6%

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

   14. associate-/l* 99.9%

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

   15. +-commutative 99.9%

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

4. Simplified 99.9%

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

   1. associate-/r/ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.3e+120) (* 0.5 (+ y (/ x (/ y x)))) (* (* z (/ z y)) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.3e+120) {
		tmp = 0.5 * (y + (x / (y / x)));
	} 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 <= 1.3d+120) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    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 <= 1.3e+120) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.3e+120:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.3e+120)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.3e+120)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.3e+120], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2999999999999999e120

   1. Initial program 70.4%

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

   2. Taylor expanded in y around 0 85.7%

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

      1. +-lft-identity 85.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 85.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 85.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 85.7%

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

      5. distribute-rgt1-in 85.7%

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

      6. distribute-lft-out 85.7%

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

      7. distribute-rgt1-in 85.7%

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

      8. metadata-eval 85.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 85.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 85.7%

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

      11. unpow2 85.7%

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

      12. unpow2 85.7%

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

      13. difference-of-squares 88.6%

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

      14. associate-/l* 99.9%

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

      15. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 66.3%

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

      1. unpow2 66.3%

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

      2. associate-/l* 73.0%

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

   7. Simplified 73.0%

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

   if 1.2999999999999999e120 < z

   1. Initial program 54.7%

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

   2. Taylor expanded in z around inf 61.9%

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

      1. *-commutative 61.9%

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

      2. unpow2 61.9%

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

      3. associate-*l/ 71.3%

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

      4. *-commutative 71.3%

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

   4. Simplified 71.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 6: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.06e+86) (* 0.5 (+ y (/ x (/ y x)))) (* 0.5 (- y (/ z (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.06e+86) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.06d+86) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = 0.5d0 * (y - (z / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.06e+86) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.06e+86:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = 0.5 * (y - (z / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.06e+86)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.06e+86)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = 0.5 * (y - (z / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.06e+86], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.06e86

   1. Initial program 70.4%

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

   2. Taylor expanded in y around 0 85.3%

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

      1. +-lft-identity 85.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 85.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 85.3%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 85.3%

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

      5. distribute-rgt1-in 85.3%

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

      6. distribute-lft-out 85.3%

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

      7. distribute-rgt1-in 85.3%

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

      8. metadata-eval 85.3%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 85.3%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 85.3%

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

      11. unpow2 85.3%

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

      12. unpow2 85.3%

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

      13. difference-of-squares 88.3%

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

      14. associate-/l* 99.9%

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

      15. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 66.3%

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

      1. unpow2 66.3%

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

      2. associate-/l* 73.2%

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

   7. Simplified 73.2%

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

   if 1.06e86 < z

   1. Initial program 56.3%

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

   2. Taylor expanded in x around 0 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. div-sub 55.0%

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

      4. associate-/l* 62.5%

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

      5. *-inverses 62.5%

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

      6. /-rgt-identity 62.5%

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

      7. associate-/l* 80.4%

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

   4. Simplified 80.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 7: 42.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e+77) (* 0.5 y) (* 0.5 (* x (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+77) {
		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 <= 4.5d+77) 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 <= 4.5e+77) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.5e+77:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e+77)
		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 <= 4.5e+77)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e+77], 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 < 4.50000000000000024e77

   1. Initial program 70.0%

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

   2. Taylor expanded in y around inf 33.9%

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

   if 4.50000000000000024e77 < x

   1. Initial program 57.1%

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

   2. Taylor expanded in x around inf 56.9%

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

      1. unpow2 56.9%

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

      2. associate-/l* 62.9%

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

   4. Simplified 62.9%

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

      1. associate-/r/ 63.0%

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

   6. Applied egg-rr 63.0%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 8: 33.1% 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 67.5%

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

2. Taylor expanded in y around inf 30.5%

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

3. Final simplification 30.5%

   \[\leadsto 0.5 \cdot y \]

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 2023297 
(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)))