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

Percentage Accurate: 69.4% → 99.9%

Time: 10.0s

Alternatives: 10

Speedup: 1.2×

• 42.7% 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: 69.4% 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

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

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (* 0.5 (+ y (* (- x_m z_m) (/ (+ x_m z_m) y)))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * (y + ((x_m - z_m) * ((x_m + z_m) / y)));
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * (y + ((x_m - z_m) * ((x_m + z_m) / y)))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * N[(y + N[(N[(x$95$m - z$95$m), $MachinePrecision] * N[(N[(x$95$m + z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

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

   1. remove-double-neg 64.6%

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

   2. distribute-lft-neg-out 64.6%

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

   3. distribute-frac-neg2 64.6%

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

   4. distribute-frac-neg 64.6%

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

   5. neg-mul-1 64.6%

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

   6. distribute-lft-neg-out 64.6%

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

   7. *-commutative 64.6%

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

   8. distribute-lft-neg-in 64.6%

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

   9. times-frac 64.6%

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

   10. metadata-eval 64.6%

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

   11. metadata-eval 64.6%

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

   12. associate--l+ 64.6%

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

   13. fma-define 66.2%

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

3. Simplified 66.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 79.0%

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

   1. associate--l+ 79.0%

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

   2. div-sub 81.7%

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

7. Simplified 81.7%

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

   1. unpow2 81.7%

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

   2. pow2 81.7%

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

   3. difference-of-squares 86.5%

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

9. Applied egg-rr 86.5%

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

    1. *-commutative 86.5%

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

    2. *-un-lft-identity 86.5%

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

    3. times-frac 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(y + z\_m \cdot \frac{x\_m - z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x\_m - z\_m\right) \cdot \frac{x\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+107)
   (* 0.5 (+ y (* z_m (/ (- x_m z_m) y))))
   (* 0.5 (+ y (* (- x_m z_m) (/ x_m y))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+107) {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	} else {
		tmp = 0.5 * (y + ((x_m - z_m) * (x_m / y)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x_m * x_m) <= 5d+107) then
        tmp = 0.5d0 * (y + (z_m * ((x_m - z_m) / y)))
    else
        tmp = 0.5d0 * (y + ((x_m - z_m) * (x_m / y)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+107) {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	} else {
		tmp = 0.5 * (y + ((x_m - z_m) * (x_m / y)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (x_m * x_m) <= 5e+107:
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)))
	else:
		tmp = 0.5 * (y + ((x_m - z_m) * (x_m / y)))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 5e+107)
		tmp = Float64(0.5 * Float64(y + Float64(z_m * Float64(Float64(x_m - z_m) / y))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x_m - z_m) * Float64(x_m / y))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 5e+107)
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	else
		tmp = 0.5 * (y + ((x_m - z_m) * (x_m / y)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 5e+107], N[(0.5 * N[(y + N[(z$95$m * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x$95$m - z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.0000000000000002e107

   1. Initial program 66.4%

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

      1. remove-double-neg 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-frac-neg2 66.4%

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

      4. distribute-frac-neg 66.4%

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

      5. neg-mul-1 66.4%

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

      6. distribute-lft-neg-out 66.4%

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

      7. *-commutative 66.4%

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

      8. distribute-lft-neg-in 66.4%

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

      9. times-frac 66.4%

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

      10. metadata-eval 66.4%

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

      11. metadata-eval 66.4%

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

      12. associate--l+ 66.4%

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

      13. fma-define 66.4%

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

   3. Simplified 66.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 91.3%

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

      1. associate--l+ 91.3%

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

      2. div-sub 91.3%

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

   7. Simplified 91.3%

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

      1. unpow2 91.3%

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

      2. pow2 91.3%

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

      3. difference-of-squares 91.3%

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

   9. Applied egg-rr 91.3%

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

       1. associate-/l* 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around 0 93.4%

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

   if 5.0000000000000002e107 < (*.f64 x x)

   1. Initial program 62.0%

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

      1. remove-double-neg 62.0%

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

      2. distribute-lft-neg-out 62.0%

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

      3. distribute-frac-neg2 62.0%

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

      4. distribute-frac-neg 62.0%

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

      5. neg-mul-1 62.0%

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

      6. distribute-lft-neg-out 62.0%

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

      7. *-commutative 62.0%

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

      8. distribute-lft-neg-in 62.0%

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

      9. times-frac 62.0%

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

      10. metadata-eval 62.0%

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

      11. metadata-eval 62.0%

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

      12. associate--l+ 62.0%

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

      13. fma-define 65.8%

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

   3. Simplified 65.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.9%

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

      1. associate--l+ 60.9%

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

      2. div-sub 67.7%

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

   7. Simplified 67.7%

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

      1. unpow2 67.7%

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

      2. pow2 67.7%

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

      3. difference-of-squares 79.4%

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

   9. Applied egg-rr 79.4%

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

   10. Taylor expanded in x around inf 71.2%

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

   11. Taylor expanded in x around 0 83.6%

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

       1. *-commutative 83.6%

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

       2. +-commutative 83.6%

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

       3. neg-mul-1 83.6%

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

       4. sub-neg 83.6%

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

       5. div-sub 84.6%

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

       6. associate-*l/ 71.2%

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

       7. associate-*r/ 84.6%

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

   13. Simplified 84.6%

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

Alternative 3: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(y + z\_m \cdot \frac{x\_m - z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x\_m - z\_m\right) \cdot \frac{x\_m + z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+71)
   (* 0.5 (+ y (* z_m (/ (- x_m z_m) y))))
   (* 0.5 (* (- x_m z_m) (/ (+ x_m z_m) y)))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+71) {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	} else {
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x_m * x_m) <= 5d+71) then
        tmp = 0.5d0 * (y + (z_m * ((x_m - z_m) / y)))
    else
        tmp = 0.5d0 * ((x_m - z_m) * ((x_m + z_m) / y))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+71) {
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	} else {
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (x_m * x_m) <= 5e+71:
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)))
	else:
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 5e+71)
		tmp = Float64(0.5 * Float64(y + Float64(z_m * Float64(Float64(x_m - z_m) / y))));
	else
		tmp = Float64(0.5 * Float64(Float64(x_m - z_m) * Float64(Float64(x_m + z_m) / y)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 5e+71)
		tmp = 0.5 * (y + (z_m * ((x_m - z_m) / y)));
	else
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 5e+71], N[(0.5 * N[(y + N[(z$95$m * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x$95$m - z$95$m), $MachinePrecision] * N[(N[(x$95$m + z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999972e71

   1. Initial program 66.5%

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

      1. remove-double-neg 66.5%

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

      2. distribute-lft-neg-out 66.5%

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

      3. distribute-frac-neg2 66.5%

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

      4. distribute-frac-neg 66.5%

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

      5. neg-mul-1 66.5%

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

      6. distribute-lft-neg-out 66.5%

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

      7. *-commutative 66.5%

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

      8. distribute-lft-neg-in 66.5%

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

      9. times-frac 66.5%

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

      10. metadata-eval 66.5%

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

      11. metadata-eval 66.5%

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

      12. associate--l+ 66.5%

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

      13. fma-define 66.5%

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

   3. Simplified 66.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 92.3%

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

      1. associate--l+ 92.3%

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

      2. div-sub 92.3%

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

   7. Simplified 92.3%

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

      1. unpow2 92.3%

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

      2. pow2 92.3%

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

      3. difference-of-squares 92.3%

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

   9. Applied egg-rr 92.3%

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

       1. associate-/l* 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around 0 94.8%

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

   if 4.99999999999999972e71 < (*.f64 x x)

   1. Initial program 62.0%

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

      1. remove-double-neg 62.0%

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

      2. distribute-lft-neg-out 62.0%

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

      3. distribute-frac-neg2 62.0%

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

      4. distribute-frac-neg 62.0%

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

      5. neg-mul-1 62.0%

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

      6. distribute-lft-neg-out 62.0%

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

      7. *-commutative 62.0%

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

      8. distribute-lft-neg-in 62.0%

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

      9. times-frac 62.0%

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

      10. metadata-eval 62.0%

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

      11. metadata-eval 62.0%

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

      12. associate--l+ 62.0%

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

      13. fma-define 65.7%

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

   3. Simplified 65.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.1%

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

      1. associate--l+ 61.1%

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

      2. div-sub 67.5%

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

   7. Simplified 67.5%

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

      1. unpow2 67.5%

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

      2. pow2 67.5%

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

      3. difference-of-squares 78.7%

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

   9. Applied egg-rr 78.7%

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

   10. Taylor expanded in y around 0 70.0%

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

       1. *-commutative 70.0%

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

       2. associate-*r/ 82.8%

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

       3. +-commutative 82.8%

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

   12. Simplified 82.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \left(y + z \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-244}:\\ \;\;\;\;\left(x\_m \cdot \frac{z\_m}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+66}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 1.05e-244)
   (* (* x_m (/ z_m y)) -0.5)
   (if (<= y 9.5e+66) (* (* x_m x_m) (/ 0.5 y)) (* 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1.05e-244) {
		tmp = (x_m * (z_m / y)) * -0.5;
	} else if (y <= 9.5e+66) {
		tmp = (x_m * x_m) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.05d-244) then
        tmp = (x_m * (z_m / y)) * (-0.5d0)
    else if (y <= 9.5d+66) then
        tmp = (x_m * x_m) * (0.5d0 / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 1.05e-244) {
		tmp = (x_m * (z_m / y)) * -0.5;
	} else if (y <= 9.5e+66) {
		tmp = (x_m * x_m) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 1.05e-244:
		tmp = (x_m * (z_m / y)) * -0.5
	elif y <= 9.5e+66:
		tmp = (x_m * x_m) * (0.5 / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 1.05e-244)
		tmp = Float64(Float64(x_m * Float64(z_m / y)) * -0.5);
	elseif (y <= 9.5e+66)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.5 / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 1.05e-244)
		tmp = (x_m * (z_m / y)) * -0.5;
	elseif (y <= 9.5e+66)
		tmp = (x_m * x_m) * (0.5 / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 1.05e-244], N[(N[(x$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 9.5e+66], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05000000000000001e-244

   1. Initial program 69.1%

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

      1. remove-double-neg 69.1%

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

      2. distribute-lft-neg-out 69.1%

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

      3. distribute-frac-neg2 69.1%

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

      4. distribute-frac-neg 69.1%

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

      5. neg-mul-1 69.1%

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

      6. distribute-lft-neg-out 69.1%

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

      7. *-commutative 69.1%

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

      8. distribute-lft-neg-in 69.1%

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

      9. times-frac 69.1%

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

      10. metadata-eval 69.1%

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

      11. metadata-eval 69.1%

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

      12. associate--l+ 69.1%

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

      13. fma-define 70.5%

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

   3. Simplified 70.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.1%

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

      1. associate--l+ 81.1%

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

      2. div-sub 83.3%

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

   7. Simplified 83.3%

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

      1. unpow2 83.3%

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

      2. pow2 83.3%

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

      3. difference-of-squares 87.7%

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

   9. Applied egg-rr 87.7%

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

   10. Taylor expanded in x around inf 60.4%

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

   11. Taylor expanded in z around inf 8.1%

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

       1. *-commutative 8.1%

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

       2. associate-/l* 10.2%

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

   13. Simplified 10.2%

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

   if 1.05000000000000001e-244 < y < 9.50000000000000051e66

   1. Initial program 86.8%

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

      1. remove-double-neg 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-frac-neg2 86.8%

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

      4. distribute-frac-neg 86.8%

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

      5. neg-mul-1 86.8%

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

      6. distribute-lft-neg-out 86.8%

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

      7. *-commutative 86.8%

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

      8. distribute-lft-neg-in 86.8%

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

      9. times-frac 86.8%

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

      10. metadata-eval 86.8%

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

      11. metadata-eval 86.8%

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

      12. associate--l+ 86.8%

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

      13. fma-define 90.1%

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

   3. Simplified 90.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-*l/ 38.0%

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

      3. associate-*r/ 37.9%

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

   7. Simplified 37.9%

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

      1. unpow2 37.9%

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

   9. Applied egg-rr 37.9%

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

   if 9.50000000000000051e66 < y

   1. Initial program 31.0%

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

      1. remove-double-neg 31.0%

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

      2. distribute-lft-neg-out 31.0%

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

      3. distribute-frac-neg2 31.0%

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

      4. distribute-frac-neg 31.0%

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

      5. neg-mul-1 31.0%

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

      6. distribute-lft-neg-out 31.0%

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

      7. *-commutative 31.0%

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

      8. distribute-lft-neg-in 31.0%

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

      9. times-frac 31.0%

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

      10. metadata-eval 31.0%

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

      11. metadata-eval 31.0%

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

      12. associate--l+ 31.0%

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

      13. fma-define 31.0%

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

   3. Simplified 31.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 65.6%

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

Alternative 5: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+168}:\\ \;\;\;\;0.5 \cdot \left(\left(x\_m - z\_m\right) \cdot \frac{x\_m + z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 3.7e+168) (* 0.5 (* (- x_m z_m) (/ (+ x_m z_m) y))) (* 0.5 y)))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.7e+168) {
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.7d+168) then
        tmp = 0.5d0 * ((x_m - z_m) * ((x_m + z_m) / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.7e+168) {
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 3.7e+168:
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 3.7e+168)
		tmp = Float64(0.5 * Float64(Float64(x_m - z_m) * Float64(Float64(x_m + z_m) / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 3.7e+168)
		tmp = 0.5 * ((x_m - z_m) * ((x_m + z_m) / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 3.7e+168], N[(0.5 * N[(N[(x$95$m - z$95$m), $MachinePrecision] * N[(N[(x$95$m + z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.70000000000000009e168

   1. Initial program 73.5%

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

      1. remove-double-neg 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-frac-neg2 73.5%

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

      4. distribute-frac-neg 73.5%

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

      5. neg-mul-1 73.5%

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

      6. distribute-lft-neg-out 73.5%

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

      7. *-commutative 73.5%

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

      8. distribute-lft-neg-in 73.5%

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

      9. times-frac 73.5%

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

      10. metadata-eval 73.5%

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

      11. metadata-eval 73.5%

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

      12. associate--l+ 73.5%

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

      13. fma-define 75.3%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.2%

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

      1. associate--l+ 81.2%

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

      2. div-sub 84.4%

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

   7. Simplified 84.4%

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

      1. unpow2 84.4%

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

      2. pow2 84.4%

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

      3. difference-of-squares 89.9%

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

   9. Applied egg-rr 89.9%

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

   10. Taylor expanded in y around 0 65.0%

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

       1. *-commutative 65.0%

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

       2. associate-*r/ 71.5%

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

       3. +-commutative 71.5%

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

   12. Simplified 71.5%

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

   if 3.70000000000000009e168 < y

   1. Initial program 10.2%

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

      1. remove-double-neg 10.2%

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

      2. distribute-lft-neg-out 10.2%

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

      3. distribute-frac-neg2 10.2%

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

      4. distribute-frac-neg 10.2%

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

      5. neg-mul-1 10.2%

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

      6. distribute-lft-neg-out 10.2%

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

      7. *-commutative 10.2%

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

      8. distribute-lft-neg-in 10.2%

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

      9. times-frac 10.2%

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

      10. metadata-eval 10.2%

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

      11. metadata-eval 10.2%

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

      12. associate--l+ 10.2%

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

      13. fma-define 10.2%

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

   3. Simplified 10.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+168}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \left(\left(x\_m - z\_m\right) \cdot \frac{x\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 3.8e+67) (* 0.5 (* (- x_m z_m) (/ x_m y))) (* 0.5 y)))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e+67) {
		tmp = 0.5 * ((x_m - z_m) * (x_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.8d+67) then
        tmp = 0.5d0 * ((x_m - z_m) * (x_m / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e+67) {
		tmp = 0.5 * ((x_m - z_m) * (x_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 3.8e+67:
		tmp = 0.5 * ((x_m - z_m) * (x_m / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 3.8e+67)
		tmp = Float64(0.5 * Float64(Float64(x_m - z_m) * Float64(x_m / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 3.8e+67)
		tmp = 0.5 * ((x_m - z_m) * (x_m / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 3.8e+67], N[(0.5 * N[(N[(x$95$m - z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000002e67

   1. Initial program 74.4%

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

      1. remove-double-neg 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-frac-neg2 74.4%

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

      4. distribute-frac-neg 74.4%

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

      5. neg-mul-1 74.4%

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

      6. distribute-lft-neg-out 74.4%

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

      7. *-commutative 74.4%

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

      8. distribute-lft-neg-in 74.4%

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

      9. times-frac 74.4%

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

      10. metadata-eval 74.4%

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

      11. metadata-eval 74.4%

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

      12. associate--l+ 74.4%

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

      13. fma-define 76.5%

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

   3. Simplified 76.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.0%

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

      1. associate--l+ 81.0%

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

      2. div-sub 84.5%

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

   7. Simplified 84.5%

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

      1. unpow2 84.5%

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

      2. pow2 84.5%

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

      3. difference-of-squares 90.7%

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

   9. Applied egg-rr 90.7%

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

   10. Taylor expanded in x around inf 60.3%

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

   11. Taylor expanded in y around 0 36.2%

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

       1. associate-*r/ 42.6%

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

       2. *-commutative 42.6%

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

       3. associate-*l/ 36.2%

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

       4. associate-*r/ 38.8%

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

   13. Simplified 38.8%

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

   if 3.8000000000000002e67 < y

   1. Initial program 31.0%

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

      1. remove-double-neg 31.0%

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

      2. distribute-lft-neg-out 31.0%

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

      3. distribute-frac-neg2 31.0%

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

      4. distribute-frac-neg 31.0%

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

      5. neg-mul-1 31.0%

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

      6. distribute-lft-neg-out 31.0%

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

      7. *-commutative 31.0%

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

      8. distribute-lft-neg-in 31.0%

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

      9. times-frac 31.0%

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

      10. metadata-eval 31.0%

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

      11. metadata-eval 31.0%

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

      12. associate--l+ 31.0%

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

      13. fma-define 31.0%

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

   3. Simplified 31.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 65.6%

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

Alternative 7: 51.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+71) (* 0.5 y) (* (* x_m x_m) (/ 0.5 y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) * (0.5 / y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x_m * x_m) <= 5d+71) then
        tmp = 0.5d0 * y
    else
        tmp = (x_m * x_m) * (0.5d0 / y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) * (0.5 / y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if (x_m * x_m) <= 5e+71:
		tmp = 0.5 * y
	else:
		tmp = (x_m * x_m) * (0.5 / y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 5e+71)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x_m * x_m) * Float64(0.5 / y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 5e+71)
		tmp = 0.5 * y;
	else
		tmp = (x_m * x_m) * (0.5 / y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 5e+71], N[(0.5 * y), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999972e71

   1. Initial program 66.5%

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

      1. remove-double-neg 66.5%

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

      2. distribute-lft-neg-out 66.5%

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

      3. distribute-frac-neg2 66.5%

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

      4. distribute-frac-neg 66.5%

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

      5. neg-mul-1 66.5%

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

      6. distribute-lft-neg-out 66.5%

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

      7. *-commutative 66.5%

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

      8. distribute-lft-neg-in 66.5%

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

      9. times-frac 66.5%

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

      10. metadata-eval 66.5%

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

      11. metadata-eval 66.5%

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

      12. associate--l+ 66.5%

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

      13. fma-define 66.5%

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

   3. Simplified 66.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 49.6%

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

   if 4.99999999999999972e71 < (*.f64 x x)

   1. Initial program 62.0%

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

      1. remove-double-neg 62.0%

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

      2. distribute-lft-neg-out 62.0%

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

      3. distribute-frac-neg2 62.0%

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

      4. distribute-frac-neg 62.0%

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

      5. neg-mul-1 62.0%

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

      6. distribute-lft-neg-out 62.0%

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

      7. *-commutative 62.0%

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

      8. distribute-lft-neg-in 62.0%

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

      9. times-frac 62.0%

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

      10. metadata-eval 62.0%

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

      11. metadata-eval 62.0%

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

      12. associate--l+ 62.0%

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

      13. fma-define 65.7%

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

   3. Simplified 65.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.6%

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

      1. *-commutative 54.6%

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

      2. associate-*l/ 54.6%

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

      3. associate-*r/ 54.6%

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

   7. Simplified 54.6%

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

      1. unpow2 54.6%

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

   9. Applied egg-rr 54.6%

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

Alternative 8: 99.8% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (* 0.5 (+ y (* (+ x_m z_m) (/ (- x_m z_m) y)))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * (y + ((x_m + z_m) * ((x_m - z_m) / y)));
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * (y + ((x_m + z_m) * ((x_m - z_m) / y)))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * N[(y + N[(N[(x$95$m + z$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

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

   1. remove-double-neg 64.6%

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

   2. distribute-lft-neg-out 64.6%

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

   3. distribute-frac-neg2 64.6%

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

   4. distribute-frac-neg 64.6%

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

   5. neg-mul-1 64.6%

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

   6. distribute-lft-neg-out 64.6%

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

   7. *-commutative 64.6%

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

   8. distribute-lft-neg-in 64.6%

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

   9. times-frac 64.6%

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

   10. metadata-eval 64.6%

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

   11. metadata-eval 64.6%

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

   12. associate--l+ 64.6%

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

   13. fma-define 66.2%

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

3. Simplified 66.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 79.0%

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

   1. associate--l+ 79.0%

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

   2. div-sub 81.7%

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

7. Simplified 81.7%

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

   1. unpow2 81.7%

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

   2. pow2 81.7%

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

   3. difference-of-squares 86.5%

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

9. Applied egg-rr 86.5%

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

    1. associate-/l* 99.9%

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

11. Applied egg-rr 99.9%

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

Alternative 9: 24.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;-0.5 \cdot \frac{x\_m \cdot z\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 3.8e-113) (* -0.5 (/ (* x_m z_m) y)) (* 0.5 y)))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-113) {
		tmp = -0.5 * ((x_m * z_m) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
real(8) function code(x_m, y, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.8d-113) then
        tmp = (-0.5d0) * ((x_m * z_m) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (y <= 3.8e-113) {
		tmp = -0.5 * ((x_m * z_m) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if y <= 3.8e-113:
		tmp = -0.5 * ((x_m * z_m) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (y <= 3.8e-113)
		tmp = Float64(-0.5 * Float64(Float64(x_m * z_m) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (y <= 3.8e-113)
		tmp = -0.5 * ((x_m * z_m) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[y, 3.8e-113], N[(-0.5 * N[(N[(x$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999983e-113

   1. Initial program 71.8%

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

      1. remove-double-neg 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-frac-neg2 71.8%

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

      4. distribute-frac-neg 71.8%

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

      5. neg-mul-1 71.8%

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

      6. distribute-lft-neg-out 71.8%

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

      7. *-commutative 71.8%

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

      8. distribute-lft-neg-in 71.8%

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

      9. times-frac 71.8%

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

      10. metadata-eval 71.8%

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

      11. metadata-eval 71.8%

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

      12. associate--l+ 71.8%

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

      13. fma-define 73.7%

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

   3. Simplified 73.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 79.8%

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

      1. associate--l+ 79.8%

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

      2. div-sub 84.2%

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

   7. Simplified 84.2%

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

      1. unpow2 84.2%

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

      2. pow2 84.2%

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

      3. difference-of-squares 89.2%

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

   9. Applied egg-rr 89.2%

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

   10. Taylor expanded in x around inf 59.3%

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

   11. Taylor expanded in z around inf 11.0%

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

   if 3.79999999999999983e-113 < y

   1. Initial program 52.2%

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

      1. remove-double-neg 52.2%

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

      2. distribute-lft-neg-out 52.2%

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

      3. distribute-frac-neg2 52.2%

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

      4. distribute-frac-neg 52.2%

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

      5. neg-mul-1 52.2%

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

      6. distribute-lft-neg-out 52.2%

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

      7. *-commutative 52.2%

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

      8. distribute-lft-neg-in 52.2%

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

      9. times-frac 52.2%

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

      10. metadata-eval 52.2%

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

      11. metadata-eval 52.2%

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

      12. associate--l+ 52.2%

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

      13. fma-define 53.2%

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

   3. Simplified 53.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 49.6%

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

Alternative 10: 33.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m) :precision binary64 (* 0.5 y))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * y

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 64.6%

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

   1. remove-double-neg 64.6%

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

   2. distribute-lft-neg-out 64.6%

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

   3. distribute-frac-neg2 64.6%

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

   4. distribute-frac-neg 64.6%

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

   5. neg-mul-1 64.6%

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

   6. distribute-lft-neg-out 64.6%

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

   7. *-commutative 64.6%

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

   8. distribute-lft-neg-in 64.6%

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

   9. times-frac 64.6%

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

   10. metadata-eval 64.6%

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

   11. metadata-eval 64.6%

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

   12. associate--l+ 64.6%

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

   13. fma-define 66.2%

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

3. Simplified 66.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 36.4%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Developer Target 1: 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 2024144 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))