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

Percentage Accurate: 70.0% → 99.9%

Time: 7.5s

Alternatives: 9

Speedup: 1.2×

• 41.0% 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: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. sub-neg 73.5%

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

   2. +-commutative 73.5%

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

   3. neg-sub0 73.5%

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

   4. associate-+l- 73.5%

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

   5. sub0-neg 73.5%

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

   6. neg-mul-1 73.5%

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

   7. *-commutative 73.5%

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

   8. times-frac 73.5%

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

   9. associate--r+ 73.5%

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

   10. div-sub 73.5%

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

   11. difference-of-squares 75.9%

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

   12. +-commutative 75.9%

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

   13. associate-*r/ 80.9%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 52.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ t_1 := z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+53}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))) (t_1 (* z (* -0.5 (/ z y)))))
   (if (<= y -1.85e-39)
     (* y 0.5)
     (if (<= y -2.8e-175)
       t_1
       (if (<= y 2.2e-89)
         t_0
         (if (<= y 1.4e-15)
           t_1
           (if (<= y 5e+53) (* y 0.5) (if (<= y 8e+82) t_0 (* y 0.5)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double t_1 = z * (-0.5 * (z / y));
	double tmp;
	if (y <= -1.85e-39) {
		tmp = y * 0.5;
	} else if (y <= -2.8e-175) {
		tmp = t_1;
	} else if (y <= 2.2e-89) {
		tmp = t_0;
	} else if (y <= 1.4e-15) {
		tmp = t_1;
	} else if (y <= 5e+53) {
		tmp = y * 0.5;
	} else if (y <= 8e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    t_1 = z * ((-0.5d0) * (z / y))
    if (y <= (-1.85d-39)) then
        tmp = y * 0.5d0
    else if (y <= (-2.8d-175)) then
        tmp = t_1
    else if (y <= 2.2d-89) then
        tmp = t_0
    else if (y <= 1.4d-15) then
        tmp = t_1
    else if (y <= 5d+53) then
        tmp = y * 0.5d0
    else if (y <= 8d+82) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double t_1 = z * (-0.5 * (z / y));
	double tmp;
	if (y <= -1.85e-39) {
		tmp = y * 0.5;
	} else if (y <= -2.8e-175) {
		tmp = t_1;
	} else if (y <= 2.2e-89) {
		tmp = t_0;
	} else if (y <= 1.4e-15) {
		tmp = t_1;
	} else if (y <= 5e+53) {
		tmp = y * 0.5;
	} else if (y <= 8e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	t_1 = z * (-0.5 * (z / y))
	tmp = 0
	if y <= -1.85e-39:
		tmp = y * 0.5
	elif y <= -2.8e-175:
		tmp = t_1
	elif y <= 2.2e-89:
		tmp = t_0
	elif y <= 1.4e-15:
		tmp = t_1
	elif y <= 5e+53:
		tmp = y * 0.5
	elif y <= 8e+82:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	t_1 = Float64(z * Float64(-0.5 * Float64(z / y)))
	tmp = 0.0
	if (y <= -1.85e-39)
		tmp = Float64(y * 0.5);
	elseif (y <= -2.8e-175)
		tmp = t_1;
	elseif (y <= 2.2e-89)
		tmp = t_0;
	elseif (y <= 1.4e-15)
		tmp = t_1;
	elseif (y <= 5e+53)
		tmp = Float64(y * 0.5);
	elseif (y <= 8e+82)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	t_1 = z * (-0.5 * (z / y));
	tmp = 0.0;
	if (y <= -1.85e-39)
		tmp = y * 0.5;
	elseif (y <= -2.8e-175)
		tmp = t_1;
	elseif (y <= 2.2e-89)
		tmp = t_0;
	elseif (y <= 1.4e-15)
		tmp = t_1;
	elseif (y <= 5e+53)
		tmp = y * 0.5;
	elseif (y <= 8e+82)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e-39], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -2.8e-175], t$95$1, If[LessEqual[y, 2.2e-89], t$95$0, If[LessEqual[y, 1.4e-15], t$95$1, If[LessEqual[y, 5e+53], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 8e+82], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85000000000000007e-39 or 1.40000000000000007e-15 < y < 5.0000000000000004e53 or 7.9999999999999997e82 < y

   1. Initial program 57.8%

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

   2. Taylor expanded in y around inf 67.6%

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

      1. *-commutative 67.6%

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

   4. Simplified 67.6%

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

   if -1.85000000000000007e-39 < y < -2.8e-175 or 2.20000000000000012e-89 < y < 1.40000000000000007e-15

   1. Initial program 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. neg-sub0 94.1%

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

      4. associate-+l- 94.1%

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

      5. sub0-neg 94.1%

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

      6. neg-mul-1 94.1%

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

      7. *-commutative 94.1%

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

      8. times-frac 94.1%

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

      9. associate--r+ 94.1%

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

      10. div-sub 94.0%

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

      11. difference-of-squares 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-*r/ 99.6%

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

      14. associate-/l* 99.6%

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

      15. *-inverses 99.6%

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

      16. /-rgt-identity 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 91.7%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. associate-*l/ 70.6%

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

      4. associate-*l* 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in z around inf 65.0%

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

      1. unpow2 65.0%

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

      2. associate-*r/ 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.0%

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

   10. Simplified 65.0%

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

   if -2.8e-175 < y < 2.20000000000000012e-89 or 5.0000000000000004e53 < y < 7.9999999999999997e82

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 62.0%

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

      1. unpow2 62.0%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in x around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. unpow2 62.0%

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

      3. metadata-eval 62.0%

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

      4. times-frac 62.0%

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

      5. associate-*r/ 61.9%

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

      6. associate-*r* 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-/r* 65.1%

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

      9. metadata-eval 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+53}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ t_1 := z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+54}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) (* x 0.5))) (t_1 (* z (* -0.5 (/ z y)))))
   (if (<= y -8.5e-36)
     (* y 0.5)
     (if (<= y -1.02e-173)
       t_1
       (if (<= y 7.6e-87)
         t_0
         (if (<= y 3.6e-17)
           t_1
           (if (<= y 5.8e+54)
             (* y 0.5)
             (if (<= y 8.2e+82) t_0 (* y 0.5)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * (x * 0.5);
	double t_1 = z * (-0.5 * (z / y));
	double tmp;
	if (y <= -8.5e-36) {
		tmp = y * 0.5;
	} else if (y <= -1.02e-173) {
		tmp = t_1;
	} else if (y <= 7.6e-87) {
		tmp = t_0;
	} else if (y <= 3.6e-17) {
		tmp = t_1;
	} else if (y <= 5.8e+54) {
		tmp = y * 0.5;
	} else if (y <= 8.2e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) * (x * 0.5d0)
    t_1 = z * ((-0.5d0) * (z / y))
    if (y <= (-8.5d-36)) then
        tmp = y * 0.5d0
    else if (y <= (-1.02d-173)) then
        tmp = t_1
    else if (y <= 7.6d-87) then
        tmp = t_0
    else if (y <= 3.6d-17) then
        tmp = t_1
    else if (y <= 5.8d+54) then
        tmp = y * 0.5d0
    else if (y <= 8.2d+82) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / y) * (x * 0.5);
	double t_1 = z * (-0.5 * (z / y));
	double tmp;
	if (y <= -8.5e-36) {
		tmp = y * 0.5;
	} else if (y <= -1.02e-173) {
		tmp = t_1;
	} else if (y <= 7.6e-87) {
		tmp = t_0;
	} else if (y <= 3.6e-17) {
		tmp = t_1;
	} else if (y <= 5.8e+54) {
		tmp = y * 0.5;
	} else if (y <= 8.2e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * (x * 0.5)
	t_1 = z * (-0.5 * (z / y))
	tmp = 0
	if y <= -8.5e-36:
		tmp = y * 0.5
	elif y <= -1.02e-173:
		tmp = t_1
	elif y <= 7.6e-87:
		tmp = t_0
	elif y <= 3.6e-17:
		tmp = t_1
	elif y <= 5.8e+54:
		tmp = y * 0.5
	elif y <= 8.2e+82:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * Float64(x * 0.5))
	t_1 = Float64(z * Float64(-0.5 * Float64(z / y)))
	tmp = 0.0
	if (y <= -8.5e-36)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.02e-173)
		tmp = t_1;
	elseif (y <= 7.6e-87)
		tmp = t_0;
	elseif (y <= 3.6e-17)
		tmp = t_1;
	elseif (y <= 5.8e+54)
		tmp = Float64(y * 0.5);
	elseif (y <= 8.2e+82)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / y) * (x * 0.5);
	t_1 = z * (-0.5 * (z / y));
	tmp = 0.0;
	if (y <= -8.5e-36)
		tmp = y * 0.5;
	elseif (y <= -1.02e-173)
		tmp = t_1;
	elseif (y <= 7.6e-87)
		tmp = t_0;
	elseif (y <= 3.6e-17)
		tmp = t_1;
	elseif (y <= 5.8e+54)
		tmp = y * 0.5;
	elseif (y <= 8.2e+82)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e-36], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.02e-173], t$95$1, If[LessEqual[y, 7.6e-87], t$95$0, If[LessEqual[y, 3.6e-17], t$95$1, If[LessEqual[y, 5.8e+54], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 8.2e+82], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000007e-36 or 3.59999999999999995e-17 < y < 5.7999999999999997e54 or 8.1999999999999999e82 < y

   1. Initial program 57.8%

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

   2. Taylor expanded in y around inf 67.6%

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

      1. *-commutative 67.6%

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

   4. Simplified 67.6%

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

   if -8.5000000000000007e-36 < y < -1.02000000000000006e-173 or 7.6e-87 < y < 3.59999999999999995e-17

   1. Initial program 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. neg-sub0 94.1%

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

      4. associate-+l- 94.1%

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

      5. sub0-neg 94.1%

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

      6. neg-mul-1 94.1%

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

      7. *-commutative 94.1%

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

      8. times-frac 94.1%

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

      9. associate--r+ 94.1%

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

      10. div-sub 94.0%

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

      11. difference-of-squares 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-*r/ 99.6%

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

      14. associate-/l* 99.6%

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

      15. *-inverses 99.6%

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

      16. /-rgt-identity 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 91.7%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. associate-*l/ 70.6%

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

      4. associate-*l* 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in z around inf 65.0%

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

      1. unpow2 65.0%

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

      2. associate-*r/ 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.0%

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

   10. Simplified 65.0%

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

   if -1.02000000000000006e-173 < y < 7.6e-87 or 5.7999999999999997e54 < y < 8.1999999999999999e82

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 62.0%

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

      1. unpow2 62.0%

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

   4. Simplified 62.0%

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

      1. times-frac 65.2%

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

      2. div-inv 65.2%

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

      3. metadata-eval 65.2%

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

   6. Applied egg-rr 65.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-173}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+54}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-174}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) (* x 0.5))))
   (if (<= y -8.5e-36)
     (* y 0.5)
     (if (<= y -1.02e-174)
       (* -0.5 (/ (* z z) y))
       (if (<= y 1.15e-91)
         t_0
         (if (<= y 2.1e-15)
           (* z (* -0.5 (/ z y)))
           (if (<= y 8.6e+54)
             (* y 0.5)
             (if (<= y 8.6e+82) t_0 (* y 0.5)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * (x * 0.5);
	double tmp;
	if (y <= -8.5e-36) {
		tmp = y * 0.5;
	} else if (y <= -1.02e-174) {
		tmp = -0.5 * ((z * z) / y);
	} else if (y <= 1.15e-91) {
		tmp = t_0;
	} else if (y <= 2.1e-15) {
		tmp = z * (-0.5 * (z / y));
	} else if (y <= 8.6e+54) {
		tmp = y * 0.5;
	} else if (y <= 8.6e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) * (x * 0.5d0)
    if (y <= (-8.5d-36)) then
        tmp = y * 0.5d0
    else if (y <= (-1.02d-174)) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (y <= 1.15d-91) then
        tmp = t_0
    else if (y <= 2.1d-15) then
        tmp = z * ((-0.5d0) * (z / y))
    else if (y <= 8.6d+54) then
        tmp = y * 0.5d0
    else if (y <= 8.6d+82) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / y) * (x * 0.5);
	double tmp;
	if (y <= -8.5e-36) {
		tmp = y * 0.5;
	} else if (y <= -1.02e-174) {
		tmp = -0.5 * ((z * z) / y);
	} else if (y <= 1.15e-91) {
		tmp = t_0;
	} else if (y <= 2.1e-15) {
		tmp = z * (-0.5 * (z / y));
	} else if (y <= 8.6e+54) {
		tmp = y * 0.5;
	} else if (y <= 8.6e+82) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * (x * 0.5)
	tmp = 0
	if y <= -8.5e-36:
		tmp = y * 0.5
	elif y <= -1.02e-174:
		tmp = -0.5 * ((z * z) / y)
	elif y <= 1.15e-91:
		tmp = t_0
	elif y <= 2.1e-15:
		tmp = z * (-0.5 * (z / y))
	elif y <= 8.6e+54:
		tmp = y * 0.5
	elif y <= 8.6e+82:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * Float64(x * 0.5))
	tmp = 0.0
	if (y <= -8.5e-36)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.02e-174)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (y <= 1.15e-91)
		tmp = t_0;
	elseif (y <= 2.1e-15)
		tmp = Float64(z * Float64(-0.5 * Float64(z / y)));
	elseif (y <= 8.6e+54)
		tmp = Float64(y * 0.5);
	elseif (y <= 8.6e+82)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / y) * (x * 0.5);
	tmp = 0.0;
	if (y <= -8.5e-36)
		tmp = y * 0.5;
	elseif (y <= -1.02e-174)
		tmp = -0.5 * ((z * z) / y);
	elseif (y <= 1.15e-91)
		tmp = t_0;
	elseif (y <= 2.1e-15)
		tmp = z * (-0.5 * (z / y));
	elseif (y <= 8.6e+54)
		tmp = y * 0.5;
	elseif (y <= 8.6e+82)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e-36], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.02e-174], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e-91], t$95$0, If[LessEqual[y, 2.1e-15], N[(z * N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+54], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 8.6e+82], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.5000000000000007e-36 or 2.09999999999999981e-15 < y < 8.59999999999999952e54 or 8.60000000000000029e82 < y

   1. Initial program 57.8%

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

   2. Taylor expanded in y around inf 67.6%

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

      1. *-commutative 67.6%

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

   4. Simplified 67.6%

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

   if -8.5000000000000007e-36 < y < -1.02000000000000011e-174

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 58.2%

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

      1. *-commutative 58.2%

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

      2. unpow2 58.2%

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

   4. Simplified 58.2%

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

   if -1.02000000000000011e-174 < y < 1.14999999999999998e-91 or 8.59999999999999952e54 < y < 8.60000000000000029e82

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 62.0%

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

      1. unpow2 62.0%

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

   4. Simplified 62.0%

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

      1. times-frac 65.2%

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

      2. div-inv 65.2%

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

      3. metadata-eval 65.2%

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

   6. Applied egg-rr 65.2%

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

   if 1.14999999999999998e-91 < y < 2.09999999999999981e-15

   1. Initial program 86.0%

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

      1. sub-neg 86.0%

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

      2. +-commutative 86.0%

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

      3. neg-sub0 86.0%

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

      4. associate-+l- 86.0%

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

      5. sub0-neg 86.0%

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

      6. neg-mul-1 86.0%

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

      7. *-commutative 86.0%

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

      8. times-frac 86.0%

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

      9. associate--r+ 86.0%

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

      10. div-sub 85.9%

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

      11. difference-of-squares 99.3%

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

      12. +-commutative 99.3%

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

      13. associate-*r/ 99.4%

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

      14. associate-/l* 99.4%

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

      15. *-inverses 99.4%

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

      16. /-rgt-identity 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around inf 92.8%

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

   5. Taylor expanded in y around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. +-commutative 74.5%

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

      3. associate-*l/ 74.6%

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

      4. associate-*l* 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in z around inf 74.5%

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

      1. unpow2 74.5%

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

      2. associate-*r/ 74.6%

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

      3. *-commutative 74.6%

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

      4. associate-*l* 74.6%

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

   10. Simplified 74.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-174}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+54}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-34}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-34)
   (* -0.5 (- (/ z (/ y z)) y))
   (* -0.5 (- (* (- z x) (/ x y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-34) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - 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) <= 1d-34) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (-0.5d0) * (((z - x) * (x / y)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-34) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-34:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-34)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-34)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-34], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999928e-35

   1. Initial program 73.0%

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

      1. sub-neg 73.0%

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

      2. +-commutative 73.0%

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

      3. neg-sub0 73.0%

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

      4. associate-+l- 73.0%

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

      5. sub0-neg 73.0%

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

      6. neg-mul-1 73.0%

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

      7. *-commutative 73.0%

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

      8. times-frac 73.0%

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

      9. associate--r+ 73.0%

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

      10. div-sub 72.9%

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

      11. difference-of-squares 72.9%

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

      12. +-commutative 72.9%

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

      13. associate-*r/ 78.3%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 87.4%

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

      1. unpow2 87.4%

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

      2. associate-/l* 94.8%

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

   8. Simplified 94.8%

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

   if 9.99999999999999928e-35 < (*.f64 x x)

   1. Initial program 74.1%

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

      1. sub-neg 74.1%

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

      2. +-commutative 74.1%

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

      3. neg-sub0 74.1%

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

      4. associate-+l- 74.1%

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

      5. sub0-neg 74.1%

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

      6. neg-mul-1 74.1%

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

      7. *-commutative 74.1%

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

      8. times-frac 74.1%

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

      9. associate--r+ 74.1%

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

      10. div-sub 74.1%

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

      11. difference-of-squares 79.0%

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

      12. +-commutative 79.0%

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

      13. associate-*l/ 83.5%

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

      14. *-commutative 83.5%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 84.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-34}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 6: 43.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0033 \lor \neg \left(x \leq 3 \cdot 10^{+29}\right) \land x \leq 9 \cdot 10^{+117}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 0.0033) (and (not (<= x 3e+29)) (<= x 9e+117)))
   (* y 0.5)
   (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 0.0033) || (!(x <= 3e+29) && (x <= 9e+117))) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / 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 <= 0.0033d0) .or. (.not. (x <= 3d+29)) .and. (x <= 9d+117)) then
        tmp = y * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 0.0033) || (!(x <= 3e+29) && (x <= 9e+117))) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 0.0033) or (not (x <= 3e+29) and (x <= 9e+117)):
		tmp = y * 0.5
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 0.0033) || (!(x <= 3e+29) && (x <= 9e+117)))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= 0.0033) || (~((x <= 3e+29)) && (x <= 9e+117)))
		tmp = y * 0.5;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 0.0033], And[N[Not[LessEqual[x, 3e+29]], $MachinePrecision], LessEqual[x, 9e+117]]], N[(y * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0033 or 2.9999999999999999e29 < x < 9e117

   1. Initial program 73.6%

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

   2. Taylor expanded in y around inf 44.7%

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

      1. *-commutative 44.7%

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

   4. Simplified 44.7%

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

   if 0.0033 < x < 2.9999999999999999e29 or 9e117 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in x around inf 67.3%

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

      1. unpow2 67.3%

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

   4. Simplified 67.3%

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

   5. Taylor expanded in x around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. metadata-eval 67.3%

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

      4. times-frac 67.3%

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

      5. associate-*r/ 67.2%

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

      6. associate-*r* 69.0%

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

      7. *-commutative 69.0%

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

      8. associate-/r* 69.0%

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

      9. metadata-eval 69.0%

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

   7. Simplified 69.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0033 \lor \neg \left(x \leq 3 \cdot 10^{+29}\right) \land x \leq 9 \cdot 10^{+117}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 7: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+236}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+236) (* -0.5 (- (/ z (/ y z)) y)) (* (/ x y) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+236) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d+236) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+236) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+236:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+236)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+236)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+236], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000005e236

   1. Initial program 73.1%

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

      1. sub-neg 73.1%

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

      2. +-commutative 73.1%

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

      3. neg-sub0 73.1%

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

      4. associate-+l- 73.1%

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

      5. sub0-neg 73.1%

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

      6. neg-mul-1 73.1%

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

      7. *-commutative 73.1%

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

      8. times-frac 73.1%

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

      9. associate--r+ 73.1%

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

      10. div-sub 73.1%

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

      11. difference-of-squares 73.1%

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

      12. +-commutative 73.1%

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

      13. associate-*r/ 77.8%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 80.3%

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

      1. unpow2 80.3%

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

      2. associate-/l* 87.5%

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

   8. Simplified 87.5%

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

   if 1.00000000000000005e236 < (*.f64 x x)

   1. Initial program 74.7%

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

   2. Taylor expanded in x around inf 80.6%

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

      1. unpow2 80.6%

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

   4. Simplified 80.6%

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

      1. times-frac 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

   6. Applied egg-rr 84.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+236}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 71.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\left(y + \frac{x \cdot x}{y}\right) \cdot \left(--0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.4e-10)
   (* (+ y (/ (* x x) y)) (- -0.5))
   (* -0.5 (- (/ z (/ y z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e-10) {
		tmp = (y + ((x * x) / y)) * -(-0.5);
	} else {
		tmp = -0.5 * ((z / (y / z)) - 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 (z <= 2.4d-10) then
        tmp = (y + ((x * x) / y)) * -(-0.5d0)
    else
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e-10) {
		tmp = (y + ((x * x) / y)) * -(-0.5);
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.4e-10:
		tmp = (y + ((x * x) / y)) * -(-0.5)
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e-10)
		tmp = Float64(Float64(y + Float64(Float64(x * x) / y)) * Float64(-(-0.5)));
	else
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.4e-10)
		tmp = (y + ((x * x) / y)) * -(-0.5);
	else
		tmp = -0.5 * ((z / (y / z)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e-10], N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * (--0.5)), $MachinePrecision], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e-10

   1. Initial program 74.8%

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

      1. sub-neg 74.8%

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

      2. +-commutative 74.8%

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

      3. neg-sub0 74.8%

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

      4. associate-+l- 74.8%

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

      5. sub0-neg 74.8%

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

      6. neg-mul-1 74.8%

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

      7. *-commutative 74.8%

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

      8. times-frac 74.8%

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

      9. associate--r+ 74.8%

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

      10. div-sub 74.7%

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

      11. difference-of-squares 77.3%

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

      12. +-commutative 77.3%

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

      13. associate-*r/ 81.3%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 76.5%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   5. Step-by-step derivation

      1. unpow2 76.5%

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

      2. associate-*r/ 76.5%

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

      3. neg-mul-1 76.5%

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

      4. distribute-rgt-neg-in 76.5%

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

   6. Simplified 76.5%

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

   if 2.4e-10 < z

   1. Initial program 69.5%

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

      1. sub-neg 69.5%

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

      2. +-commutative 69.5%

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

      3. neg-sub0 69.5%

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

      4. associate-+l- 69.5%

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

      5. sub0-neg 69.5%

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

      6. neg-mul-1 69.5%

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

      7. *-commutative 69.5%

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

      8. times-frac 69.5%

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

      9. associate--r+ 69.5%

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

      10. div-sub 69.5%

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

      11. difference-of-squares 71.3%

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

      12. +-commutative 71.3%

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

      13. associate-*r/ 79.6%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 70.7%

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

      1. unpow2 70.7%

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

      2. associate-/l* 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\left(y + \frac{x \cdot x}{y}\right) \cdot \left(--0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 9: 35.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in y around inf 40.9%

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

   1. *-commutative 40.9%

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

4. Simplified 40.9%

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

5. Final simplification 40.9%

   \[\leadsto y \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023224 
(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)))