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

Percentage Accurate: 69.0% → 99.9%

Time: 6.9s

Alternatives: 9

Speedup: 1.2×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (((z - x) * ((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 = (((z - x) * ((z + x) / y)) - y) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. sub-neg 70.1%

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

   2. +-commutative 70.1%

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

   3. neg-sub0 70.1%

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

   4. associate-+l- 70.1%

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

   5. sub0-neg 70.1%

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

   6. neg-mul-1 70.1%

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

   7. *-commutative 70.1%

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

   8. times-frac 70.1%

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

   9. associate--r+ 70.1%

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

   10. div-sub 70.0%

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

   11. difference-of-squares 75.5%

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

   12. +-commutative 75.5%

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

   13. associate-*l/ 77.3%

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

   14. *-commutative 77.3%

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

   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. Final simplification 99.9%

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

Alternative 2: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ t_1 := \frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-169}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+173}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))) (t_1 (* (/ x y) (* x 0.5))))
   (if (<= (* z z) 5e-169)
     (* y 0.5)
     (if (<= (* z z) 5e+60)
       t_1
       (if (<= (* z z) 5e+88)
         t_0
         (if (<= (* z z) 2e+121)
           t_1
           (if (<= (* z z) 4e+173) (* y 0.5) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = (x / y) * (x * 0.5);
	double tmp;
	if ((z * z) <= 5e-169) {
		tmp = y * 0.5;
	} else if ((z * z) <= 5e+60) {
		tmp = t_1;
	} else if ((z * z) <= 5e+88) {
		tmp = t_0;
	} else if ((z * z) <= 2e+121) {
		tmp = t_1;
	} else if ((z * z) <= 4e+173) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    t_1 = (x / y) * (x * 0.5d0)
    if ((z * z) <= 5d-169) then
        tmp = y * 0.5d0
    else if ((z * z) <= 5d+60) then
        tmp = t_1
    else if ((z * z) <= 5d+88) then
        tmp = t_0
    else if ((z * z) <= 2d+121) then
        tmp = t_1
    else if ((z * z) <= 4d+173) then
        tmp = y * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = (x / y) * (x * 0.5);
	double tmp;
	if ((z * z) <= 5e-169) {
		tmp = y * 0.5;
	} else if ((z * z) <= 5e+60) {
		tmp = t_1;
	} else if ((z * z) <= 5e+88) {
		tmp = t_0;
	} else if ((z * z) <= 2e+121) {
		tmp = t_1;
	} else if ((z * z) <= 4e+173) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	t_1 = (x / y) * (x * 0.5)
	tmp = 0
	if (z * z) <= 5e-169:
		tmp = y * 0.5
	elif (z * z) <= 5e+60:
		tmp = t_1
	elif (z * z) <= 5e+88:
		tmp = t_0
	elif (z * z) <= 2e+121:
		tmp = t_1
	elif (z * z) <= 4e+173:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	t_1 = Float64(Float64(x / y) * Float64(x * 0.5))
	tmp = 0.0
	if (Float64(z * z) <= 5e-169)
		tmp = Float64(y * 0.5);
	elseif (Float64(z * z) <= 5e+60)
		tmp = t_1;
	elseif (Float64(z * z) <= 5e+88)
		tmp = t_0;
	elseif (Float64(z * z) <= 2e+121)
		tmp = t_1;
	elseif (Float64(z * z) <= 4e+173)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	t_1 = (x / y) * (x * 0.5);
	tmp = 0.0;
	if ((z * z) <= 5e-169)
		tmp = y * 0.5;
	elseif ((z * z) <= 5e+60)
		tmp = t_1;
	elseif ((z * z) <= 5e+88)
		tmp = t_0;
	elseif ((z * z) <= 2e+121)
		tmp = t_1;
	elseif ((z * z) <= 4e+173)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e-169], N[(y * 0.5), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+60], t$95$1, If[LessEqual[N[(z * z), $MachinePrecision], 5e+88], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 2e+121], t$95$1, If[LessEqual[N[(z * z), $MachinePrecision], 4e+173], N[(y * 0.5), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.0000000000000002e-169 or 2.00000000000000007e121 < (*.f64 z z) < 4.0000000000000001e173

   1. Initial program 66.2%

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

   2. Taylor expanded in y around inf 53.2%

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

      1. *-commutative 53.2%

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

   4. Simplified 53.2%

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

   if 5.0000000000000002e-169 < (*.f64 z z) < 4.99999999999999975e60 or 4.99999999999999997e88 < (*.f64 z z) < 2.00000000000000007e121

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 54.2%

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

      1. unpow2 54.2%

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

   4. Simplified 54.2%

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

      1. times-frac 55.9%

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

      2. div-inv 55.9%

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

      3. metadata-eval 55.9%

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

   6. Applied egg-rr 55.9%

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

   if 4.99999999999999975e60 < (*.f64 z z) < 4.99999999999999997e88 or 4.0000000000000001e173 < (*.f64 z z)

   1. Initial program 62.5%

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

   2. Taylor expanded in z around inf 72.3%

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

      1. *-commutative 72.3%

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

      2. unpow2 72.3%

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

      3. associate-/l* 75.0%

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

   4. Simplified 75.0%

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

      1. associate-/r/ 75.1%

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

   6. Applied egg-rr 75.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-169}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+88}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+173}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 3: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-169}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+173}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= (* z z) 5e-169)
     (* y 0.5)
     (if (<= (* z z) 5e+60)
       (* (/ x y) (* x 0.5))
       (if (<= (* z z) 5e+88)
         t_0
         (if (<= (* z z) 2e+121)
           (/ x (* y (/ 2.0 x)))
           (if (<= (* z z) 4e+173) (* y 0.5) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if ((z * z) <= 5e-169) {
		tmp = y * 0.5;
	} else if ((z * z) <= 5e+60) {
		tmp = (x / y) * (x * 0.5);
	} else if ((z * z) <= 5e+88) {
		tmp = t_0;
	} else if ((z * z) <= 2e+121) {
		tmp = x / (y * (2.0 / x));
	} else if ((z * z) <= 4e+173) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    if ((z * z) <= 5d-169) then
        tmp = y * 0.5d0
    else if ((z * z) <= 5d+60) then
        tmp = (x / y) * (x * 0.5d0)
    else if ((z * z) <= 5d+88) then
        tmp = t_0
    else if ((z * z) <= 2d+121) then
        tmp = x / (y * (2.0d0 / x))
    else if ((z * z) <= 4d+173) then
        tmp = y * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if ((z * z) <= 5e-169) {
		tmp = y * 0.5;
	} else if ((z * z) <= 5e+60) {
		tmp = (x / y) * (x * 0.5);
	} else if ((z * z) <= 5e+88) {
		tmp = t_0;
	} else if ((z * z) <= 2e+121) {
		tmp = x / (y * (2.0 / x));
	} else if ((z * z) <= 4e+173) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if (z * z) <= 5e-169:
		tmp = y * 0.5
	elif (z * z) <= 5e+60:
		tmp = (x / y) * (x * 0.5)
	elif (z * z) <= 5e+88:
		tmp = t_0
	elif (z * z) <= 2e+121:
		tmp = x / (y * (2.0 / x))
	elif (z * z) <= 4e+173:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (Float64(z * z) <= 5e-169)
		tmp = Float64(y * 0.5);
	elseif (Float64(z * z) <= 5e+60)
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	elseif (Float64(z * z) <= 5e+88)
		tmp = t_0;
	elseif (Float64(z * z) <= 2e+121)
		tmp = Float64(x / Float64(y * Float64(2.0 / x)));
	elseif (Float64(z * z) <= 4e+173)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if ((z * z) <= 5e-169)
		tmp = y * 0.5;
	elseif ((z * z) <= 5e+60)
		tmp = (x / y) * (x * 0.5);
	elseif ((z * z) <= 5e+88)
		tmp = t_0;
	elseif ((z * z) <= 2e+121)
		tmp = x / (y * (2.0 / x));
	elseif ((z * z) <= 4e+173)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e-169], N[(y * 0.5), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+60], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+88], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 2e+121], N[(x / N[(y * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e+173], N[(y * 0.5), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 5.0000000000000002e-169 or 2.00000000000000007e121 < (*.f64 z z) < 4.0000000000000001e173

   1. Initial program 66.2%

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

   2. Taylor expanded in y around inf 53.2%

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

      1. *-commutative 53.2%

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

   4. Simplified 53.2%

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

   if 5.0000000000000002e-169 < (*.f64 z z) < 4.99999999999999975e60

   1. Initial program 89.2%

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

   2. Taylor expanded in x around inf 52.6%

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

      1. unpow2 52.6%

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

   4. Simplified 52.6%

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

      1. times-frac 54.4%

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

      2. div-inv 54.4%

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

      3. metadata-eval 54.4%

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

   6. Applied egg-rr 54.4%

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

   if 4.99999999999999975e60 < (*.f64 z z) < 4.99999999999999997e88 or 4.0000000000000001e173 < (*.f64 z z)

   1. Initial program 62.5%

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

   2. Taylor expanded in z around inf 72.3%

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

      1. *-commutative 72.3%

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

      2. unpow2 72.3%

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

      3. associate-/l* 75.0%

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

   4. Simplified 75.0%

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

      1. associate-/r/ 75.1%

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

   6. Applied egg-rr 75.1%

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

   if 4.99999999999999997e88 < (*.f64 z z) < 2.00000000000000007e121

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 75.3%

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

      1. unpow2 75.3%

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

   4. Simplified 75.3%

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

      1. times-frac 75.3%

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

      2. div-inv 75.3%

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

      3. metadata-eval 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. metadata-eval 75.3%

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

      2. div-inv 75.3%

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

      3. *-commutative 75.3%

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

      4. clear-num 75.3%

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

      5. frac-times 75.7%

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

      6. *-un-lft-identity 75.7%

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

   8. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{2}{x} \cdot y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-169}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+88}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+173}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+60)
   (* -0.5 (- (* (- z x) (/ x y)) y))
   (* -0.5 (- (/ z (/ y z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	} 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 * z) <= 2d+60) then
        tmp = (-0.5d0) * (((z - x) * (x / y)) - y)
    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 * z) <= 2e+60) {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+60:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+60)
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	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 * z) <= 2e+60)
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	else
		tmp = -0.5 * ((z / (y / z)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+60], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e60

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. +-commutative 75.7%

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

      3. neg-sub0 75.7%

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

      4. associate-+l- 75.7%

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

      5. sub0-neg 75.7%

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

      6. neg-mul-1 75.7%

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

      7. *-commutative 75.7%

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

      8. times-frac 75.7%

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

      9. associate--r+ 75.7%

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

      10. div-sub 75.7%

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

      11. difference-of-squares 75.7%

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

      12. +-commutative 75.7%

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

      13. associate-*l/ 78.4%

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

      14. *-commutative 78.4%

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

      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 93.1%

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

   if 1.9999999999999999e60 < (*.f64 z z)

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. +-commutative 63.0%

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

      3. neg-sub0 63.0%

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

      4. associate-+l- 63.0%

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

      5. sub0-neg 63.0%

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

      6. neg-mul-1 63.0%

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

      7. *-commutative 63.0%

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

      8. times-frac 63.0%

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

      9. associate--r+ 63.0%

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

      10. div-sub 63.0%

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

      11. difference-of-squares 75.3%

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

      12. +-commutative 75.3%

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

      13. associate-*l/ 75.9%

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

      14. *-commutative 75.9%

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

      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 z around inf 79.3%

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

      1. unpow2 79.3%

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

      2. associate-/l* 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 5: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{z}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+60)
   (* -0.5 (- (* (- z x) (/ x y)) y))
   (* -0.5 (- (* (- z x) (/ z y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (z / 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 ((z * z) <= 2d+60) then
        tmp = (-0.5d0) * (((z - x) * (x / y)) - y)
    else
        tmp = (-0.5d0) * (((z - x) * (z / y)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (z / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+60:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	else:
		tmp = -0.5 * (((z - x) * (z / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+60)
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(z / y)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+60)
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	else
		tmp = -0.5 * (((z - x) * (z / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+60], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e60

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. +-commutative 75.7%

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

      3. neg-sub0 75.7%

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

      4. associate-+l- 75.7%

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

      5. sub0-neg 75.7%

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

      6. neg-mul-1 75.7%

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

      7. *-commutative 75.7%

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

      8. times-frac 75.7%

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

      9. associate--r+ 75.7%

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

      10. div-sub 75.7%

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

      11. difference-of-squares 75.7%

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

      12. +-commutative 75.7%

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

      13. associate-*l/ 78.4%

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

      14. *-commutative 78.4%

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

      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 93.1%

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

   if 1.9999999999999999e60 < (*.f64 z z)

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. +-commutative 63.0%

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

      3. neg-sub0 63.0%

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

      4. associate-+l- 63.0%

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

      5. sub0-neg 63.0%

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

      6. neg-mul-1 63.0%

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

      7. *-commutative 63.0%

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

      8. times-frac 63.0%

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

      9. associate--r+ 63.0%

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

      10. div-sub 63.0%

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

      11. difference-of-squares 75.3%

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

      12. +-commutative 75.3%

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

      13. associate-*l/ 75.9%

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

      14. *-commutative 75.9%

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

      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 0 92.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \frac{-x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+60)
   (* -0.5 (- (* x (/ (- x) y)) y))
   (* -0.5 (- (/ z (/ y z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+60) {
		tmp = -0.5 * ((x * (-x / y)) - y);
	} 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 * z) <= 2d+60) then
        tmp = (-0.5d0) * ((x * (-x / y)) - y)
    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 * z) <= 2e+60) {
		tmp = -0.5 * ((x * (-x / y)) - y);
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+60:
		tmp = -0.5 * ((x * (-x / y)) - y)
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+60)
		tmp = Float64(-0.5 * Float64(Float64(x * Float64(Float64(-x) / y)) - y));
	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 * z) <= 2e+60)
		tmp = -0.5 * ((x * (-x / y)) - y);
	else
		tmp = -0.5 * ((z / (y / z)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+60], N[(-0.5 * N[(N[(x * N[((-x) / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e60

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. +-commutative 75.7%

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

      3. neg-sub0 75.7%

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

      4. associate-+l- 75.7%

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

      5. sub0-neg 75.7%

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

      6. neg-mul-1 75.7%

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

      7. *-commutative 75.7%

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

      8. times-frac 75.7%

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

      9. associate--r+ 75.7%

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

      10. div-sub 75.7%

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

      11. difference-of-squares 75.7%

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

      12. +-commutative 75.7%

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

      13. associate-*l/ 78.4%

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

      14. *-commutative 78.4%

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

      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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 86.5%

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

      1. unpow2 86.5%

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

      2. associate-*r/ 86.5%

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

      3. mul-1-neg 86.5%

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

      4. distribute-rgt-neg-out 86.5%

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

      5. associate-*l/ 93.0%

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

   8. Simplified 93.0%

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

   if 1.9999999999999999e60 < (*.f64 z z)

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. +-commutative 63.0%

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

      3. neg-sub0 63.0%

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

      4. associate-+l- 63.0%

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

      5. sub0-neg 63.0%

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

      6. neg-mul-1 63.0%

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

      7. *-commutative 63.0%

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

      8. times-frac 63.0%

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

      9. associate--r+ 63.0%

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

      10. div-sub 63.0%

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

      11. difference-of-squares 75.3%

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

      12. +-commutative 75.3%

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

      13. associate-*l/ 75.9%

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

      14. *-commutative 75.9%

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

      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 z around inf 79.3%

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

      1. unpow2 79.3%

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

      2. associate-/l* 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+60}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \frac{-x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 7: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+219}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+78}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.75e+219)
   (* (/ x y) (* x 0.5))
   (if (<= x 4e+78) (* -0.5 (- (/ z (/ y z)) y)) (/ x (* y (/ 2.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e+219) {
		tmp = (x / y) * (x * 0.5);
	} else if (x <= 4e+78) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = x / (y * (2.0 / 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 <= (-2.75d+219)) then
        tmp = (x / y) * (x * 0.5d0)
    else if (x <= 4d+78) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = x / (y * (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e+219) {
		tmp = (x / y) * (x * 0.5);
	} else if (x <= 4e+78) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = x / (y * (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.75e+219:
		tmp = (x / y) * (x * 0.5)
	elif x <= 4e+78:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = x / (y * (2.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.75e+219)
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	elseif (x <= 4e+78)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(x / Float64(y * Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.75e+219)
		tmp = (x / y) * (x * 0.5);
	elseif (x <= 4e+78)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = x / (y * (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.75e+219], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e+78], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.74999999999999986e219

   1. Initial program 80.1%

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

   2. Taylor expanded in x around inf 87.2%

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

      1. unpow2 87.2%

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

   4. Simplified 87.2%

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

      1. times-frac 93.4%

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

      2. div-inv 93.4%

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

      3. metadata-eval 93.4%

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

   6. Applied egg-rr 93.4%

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

   if -2.74999999999999986e219 < x < 4.00000000000000003e78

   1. Initial program 72.6%

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

      1. sub-neg 72.6%

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

      2. +-commutative 72.6%

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

      3. neg-sub0 72.6%

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

      4. associate-+l- 72.6%

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

      5. sub0-neg 72.6%

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

      6. neg-mul-1 72.6%

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

      7. *-commutative 72.6%

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

      8. times-frac 72.6%

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

      9. associate--r+ 72.6%

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

      10. div-sub 72.6%

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

      11. difference-of-squares 74.1%

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

      12. +-commutative 74.1%

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

      13. associate-*l/ 76.0%

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

      14. *-commutative 76.0%

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

      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 z around inf 76.2%

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

      1. unpow2 76.2%

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

      2. associate-/l* 81.8%

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

   6. Simplified 81.8%

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

   if 4.00000000000000003e78 < x

   1. Initial program 57.5%

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

   2. Taylor expanded in x around inf 63.4%

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

      1. unpow2 63.4%

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

   4. Simplified 63.4%

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

      1. times-frac 68.8%

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

      2. div-inv 68.8%

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

      3. metadata-eval 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. metadata-eval 68.8%

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

      2. div-inv 68.8%

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

      3. *-commutative 68.8%

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

      4. clear-num 68.8%

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

      5. frac-times 68.8%

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

      6. *-un-lft-identity 68.8%

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

   8. Applied egg-rr 68.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+219}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+78}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \]

Alternative 8: 52.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+45} \lor \neg \left(y \leq 63000000000\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+45) (not (<= y 63000000000.0)))
   (* y 0.5)
   (* -0.5 (* z (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+45) || !(y <= 63000000000.0)) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (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 ((y <= (-2.8d+45)) .or. (.not. (y <= 63000000000.0d0))) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+45) || !(y <= 63000000000.0)) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.8e+45) or not (y <= 63000000000.0):
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+45) || !(y <= 63000000000.0))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.8e+45) || ~((y <= 63000000000.0)))
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+45], N[Not[LessEqual[y, 63000000000.0]], $MachinePrecision]], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999999e45 or 6.3e10 < y

   1. Initial program 44.8%

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

   2. Taylor expanded in y around inf 67.1%

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

      1. *-commutative 67.1%

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

   4. Simplified 67.1%

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

   if -2.7999999999999999e45 < y < 6.3e10

   1. Initial program 90.6%

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

   2. Taylor expanded in z around inf 48.6%

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

      1. *-commutative 48.6%

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

      2. unpow2 48.6%

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

      3. associate-/l* 49.2%

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

   4. Simplified 49.2%

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

      1. associate-/r/ 49.3%

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

   6. Applied egg-rr 49.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+45} \lor \neg \left(y \leq 63000000000\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

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 70.1%

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

2. Taylor expanded in y around inf 35.1%

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

   1. *-commutative 35.1%

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

4. Simplified 35.1%

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

5. Final simplification 35.1%

   \[\leadsto y \cdot 0.5 \]

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 2023196 
(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)))