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

Percentage Accurate: 68.2% → 99.9%

Time: 6.4s

Alternatives: 9

Speedup: 1.2×

• 41.6% 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: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

   1. sub-neg 73.2%

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

   2. +-commutative 73.2%

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

   3. neg-sub0 73.2%

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

   4. associate-+l- 73.2%

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

   5. sub0-neg 73.2%

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

   6. neg-mul-1 73.2%

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

   7. *-commutative 73.2%

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

   8. times-frac 73.2%

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

   9. associate--r+ 73.2%

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

   10. div-sub 73.2%

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

   11. difference-of-squares 77.5%

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

   12. +-commutative 77.5%

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

   13. associate-*r/ 79.7%

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

   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. Final simplification 99.9%

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

Alternative 2: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \left(-\frac{x}{y}\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 4e+121)
   (* -0.5 (- (/ z (/ y z)) y))
   (* -0.5 (- (* (+ x z) (- (/ x y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e+121) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * (((x + z) * -(x / y)) - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 4d+121) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (-0.5d0) * (((x + z) * -(x / y)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 4e+121) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * (((x + z) * -(x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 4e+121:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = -0.5 * (((x + z) * -(x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 4e+121)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(x + z) * Float64(-Float64(x / y))) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 4e+121)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = -0.5 * (((x + z) * -(x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e+121], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(x + z), $MachinePrecision] * (-N[(x / y), $MachinePrecision])), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000015e121

   1. Initial program 75.6%

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

      1. sub-neg 75.6%

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

      2. +-commutative 75.6%

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

      3. neg-sub0 75.6%

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

      4. associate-+l- 75.6%

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

      5. sub0-neg 75.6%

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

      6. neg-mul-1 75.6%

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

      7. *-commutative 75.6%

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

      8. times-frac 75.6%

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

      9. associate--r+ 75.6%

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

      10. div-sub 75.6%

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

      11. difference-of-squares 75.6%

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

      12. +-commutative 75.6%

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

      13. associate-*r/ 76.9%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 88.0%

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

      1. unpow2 88.0%

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

      2. associate-/l* 91.1%

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

   6. Simplified 91.1%

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

   if 4.00000000000000015e121 < (*.f64 x x)

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. +-commutative 69.7%

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

      3. neg-sub0 69.7%

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

      4. associate-+l- 69.7%

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

      5. sub0-neg 69.7%

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

      6. neg-mul-1 69.7%

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

      7. *-commutative 69.7%

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

      8. times-frac 69.7%

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

      9. associate--r+ 69.7%

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

      10. div-sub 69.7%

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

      11. difference-of-squares 80.3%

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

      12. +-commutative 80.3%

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

      13. associate-*r/ 83.8%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 92.2%

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

      1. neg-mul-1 92.2%

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

      2. distribute-neg-frac 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 91.5%

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

Alternative 3: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+161}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+161)
   (* -0.5 (- (* x (- (/ x y))) y))
   (* -0.5 (- (* (+ x z) (/ z y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+161) {
		tmp = -0.5 * ((x * -(x / y)) - y);
	} else {
		tmp = -0.5 * (((x + z) * (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) <= 1d+161) then
        tmp = (-0.5d0) * ((x * -(x / y)) - y)
    else
        tmp = (-0.5d0) * (((x + z) * (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) <= 1e+161) {
		tmp = -0.5 * ((x * -(x / y)) - y);
	} else {
		tmp = -0.5 * (((x + z) * (z / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+161:
		tmp = -0.5 * ((x * -(x / y)) - y)
	else:
		tmp = -0.5 * (((x + z) * (z / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+161)
		tmp = Float64(-0.5 * Float64(Float64(x * Float64(-Float64(x / y))) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(x + z) * Float64(z / y)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+161)
		tmp = -0.5 * ((x * -(x / y)) - y);
	else
		tmp = -0.5 * (((x + z) * (z / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+161], N[(-0.5 * N[(N[(x * (-N[(x / y), $MachinePrecision])), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(x + z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e161

   1. Initial program 74.7%

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

      1. sub-neg 74.7%

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

      2. +-commutative 74.7%

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

      3. neg-sub0 74.7%

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

      4. associate-+l- 74.7%

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

      5. sub0-neg 74.7%

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

      6. neg-mul-1 74.7%

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

      7. *-commutative 74.7%

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

      8. times-frac 74.7%

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

      9. associate--r+ 74.7%

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

      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/ 76.8%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. 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 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unpow2 83.7%

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

      3. associate-*r/ 90.9%

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

      4. distribute-lft-neg-in 90.9%

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

   8. Simplified 90.9%

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

   if 1e161 < (*.f64 z z)

   1. Initial program 70.7%

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

      1. sub-neg 70.7%

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

      2. +-commutative 70.7%

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

      3. neg-sub0 70.7%

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

      4. associate-+l- 70.7%

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

      5. sub0-neg 70.7%

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

      6. neg-mul-1 70.7%

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

      7. *-commutative 70.7%

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

      8. times-frac 70.7%

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

      9. associate--r+ 70.7%

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

      10. div-sub 70.7%

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

      11. difference-of-squares 82.5%

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

      12. +-commutative 82.5%

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

      13. associate-*r/ 84.6%

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

      14. associate-/l* 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. Taylor expanded in z around inf 88.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+161}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+125}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e+125)
   (* -0.5 (- (/ z (/ y z)) y))
   (* -0.5 (- (* x (- (/ x y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+125) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * ((x * -(x / y)) - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d+125) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (-0.5d0) * ((x * -(x / y)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+125) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * ((x * -(x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e+125:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = -0.5 * ((x * -(x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+125)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(x * Float64(-Float64(x / y))) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e+125)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = -0.5 * ((x * -(x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+125], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(x * (-N[(x / y), $MachinePrecision])), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999962e125

   1. Initial program 75.8%

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

      1. sub-neg 75.8%

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

      2. +-commutative 75.8%

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

      3. neg-sub0 75.8%

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

      4. associate-+l- 75.8%

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

      5. sub0-neg 75.8%

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

      6. neg-mul-1 75.8%

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

      7. *-commutative 75.8%

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

      8. times-frac 75.8%

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

      9. associate--r+ 75.8%

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

      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-*r/ 77.0%

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

      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. Taylor expanded in x around 0 88.1%

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

      1. unpow2 88.1%

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

      2. associate-/l* 91.1%

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

   6. Simplified 91.1%

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

   if 4.99999999999999962e125 < (*.f64 x x)

   1. Initial program 69.4%

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

      1. sub-neg 69.4%

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

      2. +-commutative 69.4%

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

      3. neg-sub0 69.4%

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

      4. associate-+l- 69.4%

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

      5. sub0-neg 69.4%

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

      6. neg-mul-1 69.4%

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

      7. *-commutative 69.4%

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

      8. times-frac 69.4%

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

      9. associate--r+ 69.4%

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

      10. div-sub 69.4%

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

      11. difference-of-squares 80.2%

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

      12. +-commutative 80.2%

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

      13. associate-*r/ 83.6%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. 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 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*r/ 87.2%

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

      4. distribute-lft-neg-in 87.2%

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

   8. Simplified 87.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+125}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \end{array} \]

Alternative 5: 52.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+62}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= x -2.8e+46)
     t_0
     (if (<= x 1.9e-112)
       (* y 0.5)
       (if (<= x 7e+62) (* -0.5 (* z (/ z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (x <= -2.8e+46) {
		tmp = t_0;
	} else if (x <= 1.9e-112) {
		tmp = y * 0.5;
	} else if (x <= 7e+62) {
		tmp = -0.5 * (z * (z / y));
	} 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 = x * (x * (0.5d0 / y))
    if (x <= (-2.8d+46)) then
        tmp = t_0
    else if (x <= 1.9d-112) then
        tmp = y * 0.5d0
    else if (x <= 7d+62) then
        tmp = (-0.5d0) * (z * (z / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (x <= -2.8e+46) {
		tmp = t_0;
	} else if (x <= 1.9e-112) {
		tmp = y * 0.5;
	} else if (x <= 7e+62) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if x <= -2.8e+46:
		tmp = t_0
	elif x <= 1.9e-112:
		tmp = y * 0.5
	elif x <= 7e+62:
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (x <= -2.8e+46)
		tmp = t_0;
	elseif (x <= 1.9e-112)
		tmp = Float64(y * 0.5);
	elseif (x <= 7e+62)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (x <= -2.8e+46)
		tmp = t_0;
	elseif (x <= 1.9e-112)
		tmp = y * 0.5;
	elseif (x <= 7e+62)
		tmp = -0.5 * (z * (z / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+46], t$95$0, If[LessEqual[x, 1.9e-112], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 7e+62], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000018e46 or 6.99999999999999967e62 < x

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf 64.3%

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

      1. unpow2 64.3%

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

   4. Simplified 64.3%

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

   5. Taylor expanded in x around 0 64.3%

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

      1. unpow2 64.3%

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

      2. associate-*r/ 64.3%

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

      3. associate-*l/ 64.2%

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

      4. *-commutative 64.2%

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

      5. associate-*r* 72.5%

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

   7. Simplified 72.5%

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

   if -2.80000000000000018e46 < x < 1.89999999999999997e-112

   1. Initial program 71.5%

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

   2. Taylor expanded in y around inf 54.4%

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

      1. *-commutative 54.4%

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

   4. Simplified 54.4%

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

   if 1.89999999999999997e-112 < x < 6.99999999999999967e62

   1. Initial program 88.2%

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

   2. Taylor expanded in z around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. unpow2 56.3%

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

      3. associate-/l* 61.8%

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

   4. Simplified 61.8%

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

      1. associate-/r/ 61.8%

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

   6. Applied egg-rr 61.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+62}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 6: 52.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+63}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+46)
   (* 0.5 (/ x (/ y x)))
   (if (<= x 2e-112)
     (* y 0.5)
     (if (<= x 4.9e+63) (* -0.5 (* z (/ z y))) (* x (* x (/ 0.5 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+46) {
		tmp = 0.5 * (x / (y / x));
	} else if (x <= 2e-112) {
		tmp = y * 0.5;
	} else if (x <= 4.9e+63) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3d+46)) then
        tmp = 0.5d0 * (x / (y / x))
    else if (x <= 2d-112) then
        tmp = y * 0.5d0
    else if (x <= 4.9d+63) then
        tmp = (-0.5d0) * (z * (z / y))
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+46) {
		tmp = 0.5 * (x / (y / x));
	} else if (x <= 2e-112) {
		tmp = y * 0.5;
	} else if (x <= 4.9e+63) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e+46:
		tmp = 0.5 * (x / (y / x))
	elif x <= 2e-112:
		tmp = y * 0.5
	elif x <= 4.9e+63:
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+46)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	elseif (x <= 2e-112)
		tmp = Float64(y * 0.5);
	elseif (x <= 4.9e+63)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e+46)
		tmp = 0.5 * (x / (y / x));
	elseif (x <= 2e-112)
		tmp = y * 0.5;
	elseif (x <= 4.9e+63)
		tmp = -0.5 * (z * (z / y));
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+46], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-112], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 4.9e+63], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.00000000000000023e46

   1. Initial program 73.5%

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

   2. Taylor expanded in x around inf 67.0%

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

      1. *-commutative 67.0%

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

      2. unpow2 67.0%

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

      3. associate-/l* 72.5%

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

   4. Simplified 72.5%

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

   if -3.00000000000000023e46 < x < 1.9999999999999999e-112

   1. Initial program 71.5%

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

   2. Taylor expanded in y around inf 54.4%

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

      1. *-commutative 54.4%

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

   4. Simplified 54.4%

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

   if 1.9999999999999999e-112 < x < 4.8999999999999997e63

   1. Initial program 88.2%

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

   2. Taylor expanded in z around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. unpow2 56.3%

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

      3. associate-/l* 61.8%

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

   4. Simplified 61.8%

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

      1. associate-/r/ 61.8%

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

   6. Applied egg-rr 61.8%

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

   if 4.8999999999999997e63 < x

   1. Initial program 66.7%

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

   2. Taylor expanded in x around inf 60.7%

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

      1. unpow2 60.7%

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

   4. Simplified 60.7%

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

   5. Taylor expanded in x around 0 60.7%

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

      1. unpow2 60.7%

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

      2. associate-*r/ 60.7%

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

      3. associate-*l/ 60.6%

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

      4. *-commutative 60.6%

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

      5. associate-*r* 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+63}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 7: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+69} \lor \neg \left(x \leq 1.2 \cdot 10^{+153}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.7e+69) (not (<= x 1.2e+153)))
   (* x (* x (/ 0.5 y)))
   (* -0.5 (- (/ z (/ y z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e+69) || !(x <= 1.2e+153)) {
		tmp = x * (x * (0.5 / 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 ((x <= (-1.7d+69)) .or. (.not. (x <= 1.2d+153))) then
        tmp = x * (x * (0.5d0 / 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 ((x <= -1.7e+69) || !(x <= 1.2e+153)) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.7e+69) or not (x <= 1.2e+153):
		tmp = x * (x * (0.5 / y))
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.7e+69) || !(x <= 1.2e+153))
		tmp = Float64(x * Float64(x * Float64(0.5 / 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 ((x <= -1.7e+69) || ~((x <= 1.2e+153)))
		tmp = x * (x * (0.5 / y));
	else
		tmp = -0.5 * ((z / (y / z)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.7e+69], N[Not[LessEqual[x, 1.2e+153]], $MachinePrecision]], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $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 x < -1.69999999999999993e69 or 1.19999999999999996e153 < x

   1. Initial program 69.0%

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

   2. Taylor expanded in x around inf 71.3%

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

      1. unpow2 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around 0 71.3%

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

      1. unpow2 71.3%

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

      2. associate-*r/ 71.3%

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

      3. associate-*l/ 71.3%

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

      4. *-commutative 71.3%

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

      5. associate-*r* 81.6%

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

   7. Simplified 81.6%

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

   if -1.69999999999999993e69 < x < 1.19999999999999996e153

   1. Initial program 75.4%

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

      1. sub-neg 75.4%

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

      2. +-commutative 75.4%

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

      3. neg-sub0 75.4%

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

      4. associate-+l- 75.4%

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

      5. sub0-neg 75.4%

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

      6. neg-mul-1 75.4%

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

      7. *-commutative 75.4%

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

      8. times-frac 75.4%

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

      9. associate--r+ 75.4%

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

      10. div-sub 75.3%

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

      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-*r/ 76.5%

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

      14. associate-/l* 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 85.1%

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

      1. unpow2 85.1%

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

      2. associate-/l* 88.4%

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

   6. Simplified 88.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+69} \lor \neg \left(x \leq 1.2 \cdot 10^{+153}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 8: 52.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+46} \lor \neg \left(x \leq 1.04 \cdot 10^{+152}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.9e+46) (not (<= x 1.04e+152)))
   (* x (* x (/ 0.5 y)))
   (* y 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.9e+46) || !(x <= 1.04e+152)) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.9d+46)) .or. (.not. (x <= 1.04d+152))) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.9e+46) || !(x <= 1.04e+152)) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.9e+46) or not (x <= 1.04e+152):
		tmp = x * (x * (0.5 / y))
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.9e+46) || !(x <= 1.04e+152))
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.9e+46) || ~((x <= 1.04e+152)))
		tmp = x * (x * (0.5 / y));
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.9e+46], N[Not[LessEqual[x, 1.04e+152]], $MachinePrecision]], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e46 or 1.04000000000000005e152 < x

   1. Initial program 70.3%

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

   2. Taylor expanded in x around inf 69.3%

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

      1. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. unpow2 69.3%

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

      2. associate-*r/ 69.3%

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

      3. associate-*l/ 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-*r* 79.1%

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

   7. Simplified 79.1%

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

   if -1.9e46 < x < 1.04000000000000005e152

   1. Initial program 74.8%

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

   2. Taylor expanded in y around inf 46.4%

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

      1. *-commutative 46.4%

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

   4. Simplified 46.4%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+46} \lor \neg \left(x \leq 1.04 \cdot 10^{+152}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9: 34.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

2. Taylor expanded in y around inf 33.9%

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

   1. *-commutative 33.9%

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

4. Simplified 33.9%

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

5. Final simplification 33.9%

   \[\leadsto y \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023181 
(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)))