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

Percentage Accurate: 68.9% → 99.9%

Time: 7.7s

Alternatives: 9

Speedup: 1.2×

• 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: 68.9% 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 69.6%

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

   1. sub-neg 69.6%

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

   2. +-commutative 69.6%

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

   3. neg-sub0 69.6%

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

   4. associate-+l- 69.6%

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

   5. sub0-neg 69.6%

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

   6. neg-mul-1 69.6%

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

   7. *-commutative 69.6%

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

   8. times-frac 69.6%

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

   9. associate--r+ 69.6%

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

   10. div-sub 69.6%

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

   11. difference-of-squares 74.7%

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

   12. +-commutative 74.7%

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

   13. associate-*r/ 76.6%

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

   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. Final simplification 99.9%

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

Alternative 2: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;z \leq 2.15 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-240}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{+35} \lor \neg \left(z \leq 3.6 \cdot 10^{+56}\right) \land z \leq 4.2 \cdot 10^{+80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))))
   (if (<= z 2.15e-277)
     t_0
     (if (<= z 6.5e-240)
       (* y 0.5)
       (if (<= z 8.3e-227)
         t_0
         (if (<= z 3.8e-139)
           (* y 0.5)
           (if (<= z 7.2e-112)
             t_0
             (if (<= z 3.3e-68)
               (* y 0.5)
               (if (<= z 3.6e-41)
                 t_0
                 (if (or (<= z 1.62e+35)
                         (and (not (<= z 3.6e+56)) (<= z 4.2e+80)))
                   (* y 0.5)
                   (* -0.5 (/ z (/ y z)))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 2.15e-277) {
		tmp = t_0;
	} else if (z <= 6.5e-240) {
		tmp = y * 0.5;
	} else if (z <= 8.3e-227) {
		tmp = t_0;
	} else if (z <= 3.8e-139) {
		tmp = y * 0.5;
	} else if (z <= 7.2e-112) {
		tmp = t_0;
	} else if (z <= 3.3e-68) {
		tmp = y * 0.5;
	} else if (z <= 3.6e-41) {
		tmp = t_0;
	} else if ((z <= 1.62e+35) || (!(z <= 3.6e+56) && (z <= 4.2e+80))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / (y / x))
    if (z <= 2.15d-277) then
        tmp = t_0
    else if (z <= 6.5d-240) then
        tmp = y * 0.5d0
    else if (z <= 8.3d-227) then
        tmp = t_0
    else if (z <= 3.8d-139) then
        tmp = y * 0.5d0
    else if (z <= 7.2d-112) then
        tmp = t_0
    else if (z <= 3.3d-68) then
        tmp = y * 0.5d0
    else if (z <= 3.6d-41) then
        tmp = t_0
    else if ((z <= 1.62d+35) .or. (.not. (z <= 3.6d+56)) .and. (z <= 4.2d+80)) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z / (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 2.15e-277) {
		tmp = t_0;
	} else if (z <= 6.5e-240) {
		tmp = y * 0.5;
	} else if (z <= 8.3e-227) {
		tmp = t_0;
	} else if (z <= 3.8e-139) {
		tmp = y * 0.5;
	} else if (z <= 7.2e-112) {
		tmp = t_0;
	} else if (z <= 3.3e-68) {
		tmp = y * 0.5;
	} else if (z <= 3.6e-41) {
		tmp = t_0;
	} else if ((z <= 1.62e+35) || (!(z <= 3.6e+56) && (z <= 4.2e+80))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z / (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	tmp = 0
	if z <= 2.15e-277:
		tmp = t_0
	elif z <= 6.5e-240:
		tmp = y * 0.5
	elif z <= 8.3e-227:
		tmp = t_0
	elif z <= 3.8e-139:
		tmp = y * 0.5
	elif z <= 7.2e-112:
		tmp = t_0
	elif z <= 3.3e-68:
		tmp = y * 0.5
	elif z <= 3.6e-41:
		tmp = t_0
	elif (z <= 1.62e+35) or (not (z <= 3.6e+56) and (z <= 4.2e+80)):
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z / (y / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (z <= 2.15e-277)
		tmp = t_0;
	elseif (z <= 6.5e-240)
		tmp = Float64(y * 0.5);
	elseif (z <= 8.3e-227)
		tmp = t_0;
	elseif (z <= 3.8e-139)
		tmp = Float64(y * 0.5);
	elseif (z <= 7.2e-112)
		tmp = t_0;
	elseif (z <= 3.3e-68)
		tmp = Float64(y * 0.5);
	elseif (z <= 3.6e-41)
		tmp = t_0;
	elseif ((z <= 1.62e+35) || (!(z <= 3.6e+56) && (z <= 4.2e+80)))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if (z <= 2.15e-277)
		tmp = t_0;
	elseif (z <= 6.5e-240)
		tmp = y * 0.5;
	elseif (z <= 8.3e-227)
		tmp = t_0;
	elseif (z <= 3.8e-139)
		tmp = y * 0.5;
	elseif (z <= 7.2e-112)
		tmp = t_0;
	elseif (z <= 3.3e-68)
		tmp = y * 0.5;
	elseif (z <= 3.6e-41)
		tmp = t_0;
	elseif ((z <= 1.62e+35) || (~((z <= 3.6e+56)) && (z <= 4.2e+80)))
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z / (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.15e-277], t$95$0, If[LessEqual[z, 6.5e-240], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 8.3e-227], t$95$0, If[LessEqual[z, 3.8e-139], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 7.2e-112], t$95$0, If[LessEqual[z, 3.3e-68], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 3.6e-41], t$95$0, If[Or[LessEqual[z, 1.62e+35], And[N[Not[LessEqual[z, 3.6e+56]], $MachinePrecision], LessEqual[z, 4.2e+80]]], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.14999999999999995e-277 or 6.50000000000000007e-240 < z < 8.2999999999999996e-227 or 3.80000000000000008e-139 < z < 7.2000000000000002e-112 or 3.2999999999999998e-68 < z < 3.6e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in y around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. difference-of-squares 61.5%

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

      4. associate-/l* 65.6%

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

      5. +-commutative 65.6%

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

   4. Simplified 65.6%

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

      1. div-inv 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in z around 0 39.2%

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

      1. unpow2 39.2%

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

      2. associate-/l* 40.7%

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

   9. Simplified 40.7%

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

   if 2.14999999999999995e-277 < z < 6.50000000000000007e-240 or 8.2999999999999996e-227 < z < 3.80000000000000008e-139 or 7.2000000000000002e-112 < z < 3.2999999999999998e-68 or 3.6e-41 < z < 1.62e35 or 3.59999999999999998e56 < z < 4.20000000000000003e80

   1. Initial program 68.4%

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

   2. Taylor expanded in y around inf 63.4%

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

      1. *-commutative 63.4%

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

   4. Simplified 63.4%

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

   if 1.62e35 < z < 3.59999999999999998e56 or 4.20000000000000003e80 < z

   1. Initial program 71.5%

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

   2. Taylor expanded in z around inf 63.2%

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

      1. unpow2 63.2%

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

      2. associate-*r/ 63.2%

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

   4. Simplified 63.2%

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

      1. frac-2neg 63.2%

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

      2. div-inv 63.2%

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

      3. *-commutative 63.2%

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

      4. distribute-rgt-neg-in 63.2%

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

      5. metadata-eval 63.2%

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

   6. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot 0.5\right) \cdot \frac{1}{-y}} \]
   7. Step-by-step derivation

      1. associate-*r/ 63.2%

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

      2. *-rgt-identity 63.2%

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

      3. metadata-eval 63.2%

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

      4. distribute-rgt-neg-in 63.2%

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

      5. associate-*r* 63.2%

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

      6. distribute-neg-frac 63.2%

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

      7. associate-/l* 67.6%

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

      8. distribute-neg-frac 67.6%

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

      9. neg-mul-1 67.6%

         \[\leadsto \frac{-z}{\frac{\color{blue}{-1 \cdot y}}{z \cdot -0.5}} \]

      10. *-commutative 67.6%

          \[\leadsto \frac{-z}{\frac{-1 \cdot y}{\color{blue}{-0.5 \cdot z}}} \]

      11. times-frac 67.6%

          \[\leadsto \frac{-z}{\color{blue}{\frac{-1}{-0.5} \cdot \frac{y}{z}}} \]

      12. metadata-eval 67.6%

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

   8. Simplified 67.6%

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

      1. neg-mul-1 67.6%

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

      2. times-frac 67.6%

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

      3. metadata-eval 67.6%

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

   10. Applied egg-rr 67.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-240}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-41}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{+35} \lor \neg \left(z \leq 3.6 \cdot 10^{+56}\right) \land z \leq 4.2 \cdot 10^{+80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternative 3: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;z \leq 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-238}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+34}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))))
   (if (<= z 1e-275)
     t_0
     (if (<= z 1.4e-238)
       (* y 0.5)
       (if (<= z 7.6e-227)
         t_0
         (if (<= z 1.32e-138)
           (* y 0.5)
           (if (<= z 2.6e-112)
             t_0
             (if (<= z 8.8e-68)
               (* y 0.5)
               (if (<= z 2.6e-41)
                 t_0
                 (if (<= z 9.2e+34)
                   (* y 0.5)
                   (if (<= z 6e+56)
                     (* (* z -0.5) (/ z y))
                     (if (<= z 3.8e+80)
                       (* y 0.5)
                       (* -0.5 (/ z (/ y z)))))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 1e-275) {
		tmp = t_0;
	} else if (z <= 1.4e-238) {
		tmp = y * 0.5;
	} else if (z <= 7.6e-227) {
		tmp = t_0;
	} else if (z <= 1.32e-138) {
		tmp = y * 0.5;
	} else if (z <= 2.6e-112) {
		tmp = t_0;
	} else if (z <= 8.8e-68) {
		tmp = y * 0.5;
	} else if (z <= 2.6e-41) {
		tmp = t_0;
	} else if (z <= 9.2e+34) {
		tmp = y * 0.5;
	} else if (z <= 6e+56) {
		tmp = (z * -0.5) * (z / y);
	} else if (z <= 3.8e+80) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x / (y / x))
    if (z <= 1d-275) then
        tmp = t_0
    else if (z <= 1.4d-238) then
        tmp = y * 0.5d0
    else if (z <= 7.6d-227) then
        tmp = t_0
    else if (z <= 1.32d-138) then
        tmp = y * 0.5d0
    else if (z <= 2.6d-112) then
        tmp = t_0
    else if (z <= 8.8d-68) then
        tmp = y * 0.5d0
    else if (z <= 2.6d-41) then
        tmp = t_0
    else if (z <= 9.2d+34) then
        tmp = y * 0.5d0
    else if (z <= 6d+56) then
        tmp = (z * (-0.5d0)) * (z / y)
    else if (z <= 3.8d+80) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z / (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (z <= 1e-275) {
		tmp = t_0;
	} else if (z <= 1.4e-238) {
		tmp = y * 0.5;
	} else if (z <= 7.6e-227) {
		tmp = t_0;
	} else if (z <= 1.32e-138) {
		tmp = y * 0.5;
	} else if (z <= 2.6e-112) {
		tmp = t_0;
	} else if (z <= 8.8e-68) {
		tmp = y * 0.5;
	} else if (z <= 2.6e-41) {
		tmp = t_0;
	} else if (z <= 9.2e+34) {
		tmp = y * 0.5;
	} else if (z <= 6e+56) {
		tmp = (z * -0.5) * (z / y);
	} else if (z <= 3.8e+80) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z / (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	tmp = 0
	if z <= 1e-275:
		tmp = t_0
	elif z <= 1.4e-238:
		tmp = y * 0.5
	elif z <= 7.6e-227:
		tmp = t_0
	elif z <= 1.32e-138:
		tmp = y * 0.5
	elif z <= 2.6e-112:
		tmp = t_0
	elif z <= 8.8e-68:
		tmp = y * 0.5
	elif z <= 2.6e-41:
		tmp = t_0
	elif z <= 9.2e+34:
		tmp = y * 0.5
	elif z <= 6e+56:
		tmp = (z * -0.5) * (z / y)
	elif z <= 3.8e+80:
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z / (y / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (z <= 1e-275)
		tmp = t_0;
	elseif (z <= 1.4e-238)
		tmp = Float64(y * 0.5);
	elseif (z <= 7.6e-227)
		tmp = t_0;
	elseif (z <= 1.32e-138)
		tmp = Float64(y * 0.5);
	elseif (z <= 2.6e-112)
		tmp = t_0;
	elseif (z <= 8.8e-68)
		tmp = Float64(y * 0.5);
	elseif (z <= 2.6e-41)
		tmp = t_0;
	elseif (z <= 9.2e+34)
		tmp = Float64(y * 0.5);
	elseif (z <= 6e+56)
		tmp = Float64(Float64(z * -0.5) * Float64(z / y));
	elseif (z <= 3.8e+80)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if (z <= 1e-275)
		tmp = t_0;
	elseif (z <= 1.4e-238)
		tmp = y * 0.5;
	elseif (z <= 7.6e-227)
		tmp = t_0;
	elseif (z <= 1.32e-138)
		tmp = y * 0.5;
	elseif (z <= 2.6e-112)
		tmp = t_0;
	elseif (z <= 8.8e-68)
		tmp = y * 0.5;
	elseif (z <= 2.6e-41)
		tmp = t_0;
	elseif (z <= 9.2e+34)
		tmp = y * 0.5;
	elseif (z <= 6e+56)
		tmp = (z * -0.5) * (z / y);
	elseif (z <= 3.8e+80)
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z / (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1e-275], t$95$0, If[LessEqual[z, 1.4e-238], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 7.6e-227], t$95$0, If[LessEqual[z, 1.32e-138], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 2.6e-112], t$95$0, If[LessEqual[z, 8.8e-68], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 2.6e-41], t$95$0, If[LessEqual[z, 9.2e+34], N[(y * 0.5), $MachinePrecision], If[LessEqual[z, 6e+56], N[(N[(z * -0.5), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+80], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 9.99999999999999934e-276 or 1.40000000000000002e-238 < z < 7.60000000000000019e-227 or 1.32e-138 < z < 2.59999999999999992e-112 or 8.80000000000000009e-68 < z < 2.5999999999999999e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in y around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. difference-of-squares 61.5%

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

      4. associate-/l* 65.6%

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

      5. +-commutative 65.6%

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

   4. Simplified 65.6%

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

      1. div-inv 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in z around 0 39.2%

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

      1. unpow2 39.2%

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

      2. associate-/l* 40.7%

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

   9. Simplified 40.7%

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

   if 9.99999999999999934e-276 < z < 1.40000000000000002e-238 or 7.60000000000000019e-227 < z < 1.32e-138 or 2.59999999999999992e-112 < z < 8.80000000000000009e-68 or 2.5999999999999999e-41 < z < 9.1999999999999993e34 or 6.00000000000000012e56 < z < 3.79999999999999997e80

   1. Initial program 68.4%

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

   2. Taylor expanded in y around inf 63.4%

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

      1. *-commutative 63.4%

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

   4. Simplified 63.4%

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

   if 9.1999999999999993e34 < z < 6.00000000000000012e56

   1. Initial program 77.2%

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

   2. Taylor expanded in z around inf 32.2%

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

      1. unpow2 32.2%

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

      2. associate-*r/ 32.2%

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

   4. Simplified 32.2%

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

      1. div-inv 32.2%

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

      2. associate-*r* 32.2%

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

      3. associate-*l* 32.2%

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

      4. div-inv 32.2%

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

   6. Applied egg-rr 32.2%

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

   if 3.79999999999999997e80 < z

   1. Initial program 71.1%

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

   2. Taylor expanded in z around inf 65.3%

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

      1. unpow2 65.3%

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

      2. associate-*r/ 65.3%

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

   4. Simplified 65.3%

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

      1. frac-2neg 65.3%

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

      2. div-inv 65.3%

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

      3. *-commutative 65.3%

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

      4. distribute-rgt-neg-in 65.3%

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

      5. metadata-eval 65.3%

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

   6. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot 0.5\right) \cdot \frac{1}{-y}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.3%

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

      2. *-rgt-identity 65.3%

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

      3. metadata-eval 65.3%

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

      4. distribute-rgt-neg-in 65.3%

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

      5. associate-*r* 65.3%

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

      6. distribute-neg-frac 65.3%

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

      7. associate-/l* 70.1%

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

      8. distribute-neg-frac 70.1%

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

      9. neg-mul-1 70.1%

         \[\leadsto \frac{-z}{\frac{\color{blue}{-1 \cdot y}}{z \cdot -0.5}} \]

      10. *-commutative 70.1%

          \[\leadsto \frac{-z}{\frac{-1 \cdot y}{\color{blue}{-0.5 \cdot z}}} \]

      11. times-frac 70.1%

          \[\leadsto \frac{-z}{\color{blue}{\frac{-1}{-0.5} \cdot \frac{y}{z}}} \]

      12. metadata-eval 70.1%

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

   8. Simplified 70.1%

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

      1. neg-mul-1 70.1%

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

      2. times-frac 70.1%

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

      3. metadata-eval 70.1%

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

   10. Applied egg-rr 70.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-275}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-238}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+34}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+80}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternative 4: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+260} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+288}\right) \land x \cdot x \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 5e+260)
         (and (not (<= (* x x) 2e+288)) (<= (* x x) 2e+304)))
   (* -0.5 (- (/ z (/ y z)) y))
   (* 0.5 (/ x (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 5e+260) || (!((x * x) <= 2e+288) && ((x * x) <= 2e+304))) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x * x) <= 5d+260) .or. (.not. ((x * x) <= 2d+288)) .and. ((x * x) <= 2d+304)) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 5e+260) || (!((x * x) <= 2e+288) && ((x * x) <= 2e+304))) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 5e+260) or (not ((x * x) <= 2e+288) and ((x * x) <= 2e+304)):
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 5e+260) || (!(Float64(x * x) <= 2e+288) && (Float64(x * x) <= 2e+304)))
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 5e+260) || (~(((x * x) <= 2e+288)) && ((x * x) <= 2e+304)))
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 5e+260], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 2e+288]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2e+304]]], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999996e260 or 2e288 < (*.f64 x x) < 1.9999999999999999e304

   1. Initial program 71.0%

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

      1. sub-neg 71.0%

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

      2. +-commutative 71.0%

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

      3. neg-sub0 71.0%

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

      4. associate-+l- 71.0%

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

      5. sub0-neg 71.0%

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

      6. neg-mul-1 71.0%

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

      7. *-commutative 71.0%

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

      8. times-frac 71.0%

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

      9. associate--r+ 71.0%

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

      10. div-sub 71.0%

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

      11. difference-of-squares 71.0%

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

      12. +-commutative 71.0%

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

      13. associate-*r/ 73.2%

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

      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 0 82.8%

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

      1. unpow2 82.8%

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

      2. associate-/l* 88.4%

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

   6. Simplified 88.4%

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

   if 4.9999999999999996e260 < (*.f64 x x) < 2e288 or 1.9999999999999999e304 < (*.f64 x x)

   1. Initial program 66.0%

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

   2. Taylor expanded in y around 0 66.0%

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

      1. unpow2 66.0%

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

      2. unpow2 66.0%

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

      3. difference-of-squares 83.6%

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

      4. associate-/l* 87.5%

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

      5. +-commutative 87.5%

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

   4. Simplified 87.5%

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

      1. div-inv 87.5%

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

   6. Applied egg-rr 87.5%

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

   7. Taylor expanded in z around 0 76.9%

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

      1. unpow2 76.9%

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

      2. associate-/l* 80.4%

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

   9. Simplified 80.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+260} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+288}\right) \land x \cdot x \leq 2 \cdot 10^{+304}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 5: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x \cdot \frac{x}{y}\right) \cdot \left(--0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-7)
   (* (+ y (* x (/ x y))) (- -0.5))
   (if (<= (* z z) 5e+300)
     (* -0.5 (- (/ z (/ y z)) y))
     (* 0.5 (* (+ x z) (/ (- x z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-7) {
		tmp = (y + (x * (x / y))) * -(-0.5);
	} else if ((z * z) <= 5e+300) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = 0.5 * ((x + z) * ((x - 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 * z) <= 2d-7) then
        tmp = (y + (x * (x / y))) * -(-0.5d0)
    else if ((z * z) <= 5d+300) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-7) {
		tmp = (y + (x * (x / y))) * -(-0.5);
	} else if ((z * z) <= 5e+300) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-7:
		tmp = (y + (x * (x / y))) * -(-0.5)
	elif (z * z) <= 5e+300:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-7)
		tmp = Float64(Float64(y + Float64(x * Float64(x / y))) * Float64(-(-0.5)));
	elseif (Float64(z * z) <= 5e+300)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-7)
		tmp = (y + (x * (x / y))) * -(-0.5);
	elseif ((z * z) <= 5e+300)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-7], N[(N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (--0.5)), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+300], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.9999999999999999e-7

   1. Initial program 74.6%

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

      1. sub-neg 74.6%

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

      2. +-commutative 74.6%

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

      3. neg-sub0 74.6%

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

      4. associate-+l- 74.6%

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

      5. sub0-neg 74.6%

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

      6. neg-mul-1 74.6%

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

      7. *-commutative 74.6%

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

      8. times-frac 74.6%

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

      9. associate--r+ 74.6%

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

      10. div-sub 74.6%

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

      11. difference-of-squares 74.6%

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

      12. +-commutative 74.6%

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

      13. associate-*r/ 76.0%

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

      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. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 94.6%

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

      4. distribute-rgt-neg-in 94.6%

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

   8. Simplified 94.6%

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

   if 1.9999999999999999e-7 < (*.f64 z z) < 5.00000000000000026e300

   1. Initial program 69.9%

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

      1. sub-neg 69.9%

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

      2. +-commutative 69.9%

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

      3. neg-sub0 69.9%

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

      4. associate-+l- 69.9%

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

      5. sub0-neg 69.9%

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

      6. neg-mul-1 69.9%

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

      7. *-commutative 69.9%

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

      8. times-frac 69.9%

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

      9. associate--r+ 69.9%

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

      10. div-sub 69.9%

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

      11. difference-of-squares 69.9%

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

      12. +-commutative 69.9%

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

      13. associate-*r/ 69.9%

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

      14. associate-/l* 99.8%

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

      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. Taylor expanded in x around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-/l* 78.3%

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

   6. Simplified 78.3%

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

   if 5.00000000000000026e300 < (*.f64 z z)

   1. Initial program 59.6%

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

   2. Taylor expanded in y around 0 63.1%

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

      1. unpow2 63.1%

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

      2. unpow2 63.1%

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

      3. difference-of-squares 83.0%

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

      4. associate-/l* 93.9%

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

      5. +-commutative 93.9%

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

   4. Simplified 93.9%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*r/ 93.9%

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

   7. Simplified 93.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x \cdot \frac{x}{y}\right) \cdot \left(--0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]

Alternative 6: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-107}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4e-107)
   (* y 0.5)
   (if (<= (* x x) 4e+92) (* (* z z) (/ -0.5 y)) (* 0.5 (/ x (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e-107) {
		tmp = y * 0.5;
	} else if ((x * x) <= 4e+92) {
		tmp = (z * z) * (-0.5 / y);
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 4d-107) then
        tmp = y * 0.5d0
    else if ((x * x) <= 4d+92) then
        tmp = (z * z) * ((-0.5d0) / y)
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e-107) {
		tmp = y * 0.5;
	} else if ((x * x) <= 4e+92) {
		tmp = (z * z) * (-0.5 / y);
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 4e-107:
		tmp = y * 0.5
	elif (x * x) <= 4e+92:
		tmp = (z * z) * (-0.5 / y)
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4e-107)
		tmp = Float64(y * 0.5);
	elseif (Float64(x * x) <= 4e+92)
		tmp = Float64(Float64(z * z) * Float64(-0.5 / y));
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 4e-107)
		tmp = y * 0.5;
	elseif ((x * x) <= 4e+92)
		tmp = (z * z) * (-0.5 / y);
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-107], N[(y * 0.5), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+92], N[(N[(z * z), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4e-107

   1. Initial program 72.9%

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

   2. Taylor expanded in y around inf 50.7%

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

      1. *-commutative 50.7%

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

   4. Simplified 50.7%

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

   if 4e-107 < (*.f64 x x) < 4.0000000000000002e92

   1. Initial program 73.0%

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

   2. Taylor expanded in z around inf 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*r/ 53.3%

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

   4. Simplified 53.3%

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

   5. Taylor expanded in z around 0 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*r/ 53.3%

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

      3. *-commutative 53.3%

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

      4. associate-*r/ 53.3%

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

   7. Simplified 53.3%

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

   if 4.0000000000000002e92 < (*.f64 x x)

   1. Initial program 64.6%

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

   2. Taylor expanded in y around 0 62.6%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. difference-of-squares 75.3%

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

      4. associate-/l* 80.8%

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

      5. +-commutative 80.8%

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

   4. Simplified 80.8%

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

      1. div-inv 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in z around 0 63.5%

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

      1. unpow2 63.5%

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

      2. associate-/l* 66.0%

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

   9. Simplified 66.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-107}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 7: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+240}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x \cdot \frac{x}{y}\right) \cdot \left(--0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+240)
   (* -0.5 (- (/ z (/ y z)) y))
   (* (+ y (* x (/ x y))) (- -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+240) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = (y + (x * (x / 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 ((x * x) <= 1d+240) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (y + (x * (x / y))) * -(-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+240) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = (y + (x * (x / y))) * -(-0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+240:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = (y + (x * (x / y))) * -(-0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+240)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(Float64(y + Float64(x * Float64(x / y))) * Float64(-(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+240)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = (y + (x * (x / y))) * -(-0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+240], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (--0.5)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000001e240

   1. Initial program 72.4%

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

      1. sub-neg 72.4%

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

      2. +-commutative 72.4%

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

      3. neg-sub0 72.4%

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

      4. associate-+l- 72.4%

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

      5. sub0-neg 72.4%

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

      6. neg-mul-1 72.4%

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

      7. *-commutative 72.4%

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

      8. times-frac 72.4%

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

      9. associate--r+ 72.4%

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

      10. div-sub 72.4%

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

      11. difference-of-squares 72.4%

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

      12. +-commutative 72.4%

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

      13. associate-*r/ 73.7%

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

      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 0 83.9%

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

      1. unpow2 83.9%

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

      2. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

   if 1.00000000000000001e240 < (*.f64 x x)

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. +-commutative 63.8%

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

      3. neg-sub0 63.8%

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

      4. associate-+l- 63.8%

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

      5. sub0-neg 63.8%

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

      6. neg-mul-1 63.8%

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

      7. *-commutative 63.8%

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

      8. times-frac 63.8%

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

      9. associate--r+ 63.8%

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

      10. div-sub 63.8%

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

      11. difference-of-squares 79.2%

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

      12. +-commutative 79.2%

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

      13. associate-*r/ 82.5%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   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. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. unpow2 75.2%

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

      3. associate-*l/ 89.0%

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

      4. distribute-rgt-neg-in 89.0%

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

   8. Simplified 89.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+240}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x \cdot \frac{x}{y}\right) \cdot \left(--0.5\right)\\ \end{array} \]

Alternative 8: 43.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{+36} \lor \neg \left(z \leq 7.5 \cdot 10^{+56}\right) \land z \leq 7.4 \cdot 10^{+81}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.35e+36) (and (not (<= z 7.5e+56)) (<= z 7.4e+81)))
   (* y 0.5)
   (* -0.5 (/ z (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.35e+36) || (!(z <= 7.5e+56) && (z <= 7.4e+81))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (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 <= 2.35d+36) .or. (.not. (z <= 7.5d+56)) .and. (z <= 7.4d+81)) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z / (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.35e+36) || (!(z <= 7.5e+56) && (z <= 7.4e+81))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z / (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2.35e+36) or (not (z <= 7.5e+56) and (z <= 7.4e+81)):
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z / (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2.35e+36) || (!(z <= 7.5e+56) && (z <= 7.4e+81)))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 2.35e+36) || (~((z <= 7.5e+56)) && (z <= 7.4e+81)))
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z / (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2.35e+36], And[N[Not[LessEqual[z, 7.5e+56]], $MachinePrecision], LessEqual[z, 7.4e+81]]], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.34999999999999994e36 or 7.4999999999999999e56 < z < 7.4000000000000001e81

   1. Initial program 68.9%

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

   2. Taylor expanded in y around inf 44.0%

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

      1. *-commutative 44.0%

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

   4. Simplified 44.0%

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

   if 2.34999999999999994e36 < z < 7.4999999999999999e56 or 7.4000000000000001e81 < z

   1. Initial program 71.5%

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

   2. Taylor expanded in z around inf 63.2%

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

      1. unpow2 63.2%

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

      2. associate-*r/ 63.2%

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

   4. Simplified 63.2%

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

      1. frac-2neg 63.2%

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

      2. div-inv 63.2%

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

      3. *-commutative 63.2%

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

      4. distribute-rgt-neg-in 63.2%

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

      5. metadata-eval 63.2%

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

   6. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot 0.5\right) \cdot \frac{1}{-y}} \]
   7. Step-by-step derivation

      1. associate-*r/ 63.2%

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

      2. *-rgt-identity 63.2%

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

      3. metadata-eval 63.2%

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

      4. distribute-rgt-neg-in 63.2%

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

      5. associate-*r* 63.2%

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

      6. distribute-neg-frac 63.2%

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

      7. associate-/l* 67.6%

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

      8. distribute-neg-frac 67.6%

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

      9. neg-mul-1 67.6%

         \[\leadsto \frac{-z}{\frac{\color{blue}{-1 \cdot y}}{z \cdot -0.5}} \]

      10. *-commutative 67.6%

          \[\leadsto \frac{-z}{\frac{-1 \cdot y}{\color{blue}{-0.5 \cdot z}}} \]

      11. times-frac 67.6%

          \[\leadsto \frac{-z}{\color{blue}{\frac{-1}{-0.5} \cdot \frac{y}{z}}} \]

      12. metadata-eval 67.6%

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

   8. Simplified 67.6%

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

      1. neg-mul-1 67.6%

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

      2. times-frac 67.6%

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

      3. metadata-eval 67.6%

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

   10. Applied egg-rr 67.6%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{+36} \lor \neg \left(z \leq 7.5 \cdot 10^{+56}\right) \land z \leq 7.4 \cdot 10^{+81}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternative 9: 33.8% 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 69.6%

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

2. Taylor expanded in y around inf 36.4%

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

   1. *-commutative 36.4%

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

4. Simplified 36.4%

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

5. Final simplification 36.4%

   \[\leadsto y \cdot 0.5 \]

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 2023230 
(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)))