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

Percentage Accurate: 69.0% → 99.9%

Time: 22.9s

Alternatives: 9

Speedup: 1.2×

• 41.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

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) y) (+ x_m z_m)))))

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) / y) * (x_m + z_m)));
}

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) / y) * (x_m + z_m)))
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) / y) * (x_m + z_m)));
}

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) / y) * (x_m + z_m)))

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return Float64(0.5 * Float64(y + Float64(Float64(Float64(x_m - z_m) / y) * Float64(x_m + z_m))))
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) / y) * (x_m + z_m)));
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[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.2%

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

   1. remove-double-neg 71.2%

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

   2. distribute-lft-neg-out 71.2%

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

   3. distribute-frac-neg2 71.2%

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

   4. distribute-frac-neg 71.2%

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

   5. neg-mul-1 71.2%

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

   6. distribute-lft-neg-out 71.2%

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

   7. *-commutative 71.2%

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

   8. distribute-lft-neg-in 71.2%

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

   9. times-frac 71.2%

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

   10. metadata-eval 71.2%

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

   11. metadata-eval 71.2%

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

   12. associate--l+ 71.2%

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

   13. fma-define 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in x around 0 77.1%

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

   1. associate--l+ 77.1%

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

   2. div-sub 83.7%

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

7. Simplified 83.7%

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

   1. pow2 83.7%

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

   2. pow2 83.7%

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

   3. difference-of-squares 89.3%

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

9. Applied egg-rr 89.3%

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

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

Alternative 2: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{x\_m - z\_m}{y} \cdot \left(x\_m + z\_m\right)\right)\\ \mathbf{if}\;y \leq 5.1 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z\_m \cdot z\_m}{y}\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ (- x_m z_m) y) (+ x_m z_m)))))
   (if (<= y 5.1e+70)
     t_0
     (if (<= y 4.4e+191)
       (* 0.5 (- y (/ (* z_m z_m) y)))
       (if (<= y 1.1e+224) t_0 (* 0.5 y))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double t_0 = 0.5 * (((x_m - z_m) / y) * (x_m + z_m));
	double tmp;
	if (y <= 5.1e+70) {
		tmp = t_0;
	} else if (y <= 4.4e+191) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else if (y <= 1.1e+224) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (((x_m - z_m) / y) * (x_m + z_m))
    if (y <= 5.1d+70) then
        tmp = t_0
    else if (y <= 4.4d+191) then
        tmp = 0.5d0 * (y - ((z_m * z_m) / y))
    else if (y <= 1.1d+224) then
        tmp = t_0
    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 t_0 = 0.5 * (((x_m - z_m) / y) * (x_m + z_m));
	double tmp;
	if (y <= 5.1e+70) {
		tmp = t_0;
	} else if (y <= 4.4e+191) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else if (y <= 1.1e+224) {
		tmp = t_0;
	} 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):
	t_0 = 0.5 * (((x_m - z_m) / y) * (x_m + z_m))
	tmp = 0
	if y <= 5.1e+70:
		tmp = t_0
	elif y <= 4.4e+191:
		tmp = 0.5 * (y - ((z_m * z_m) / y))
	elif y <= 1.1e+224:
		tmp = t_0
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	t_0 = Float64(0.5 * Float64(Float64(Float64(x_m - z_m) / y) * Float64(x_m + z_m)))
	tmp = 0.0
	if (y <= 5.1e+70)
		tmp = t_0;
	elseif (y <= 4.4e+191)
		tmp = Float64(0.5 * Float64(y - Float64(Float64(z_m * z_m) / y)));
	elseif (y <= 1.1e+224)
		tmp = t_0;
	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)
	t_0 = 0.5 * (((x_m - z_m) / y) * (x_m + z_m));
	tmp = 0.0;
	if (y <= 5.1e+70)
		tmp = t_0;
	elseif (y <= 4.4e+191)
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	elseif (y <= 1.1e+224)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.1e+70], t$95$0, If[LessEqual[y, 4.4e+191], N[(0.5 * N[(y - N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+224], t$95$0, N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.10000000000000014e70 or 4.4e191 < y < 1.1e224

   1. Initial program 77.3%

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

      1. remove-double-neg 77.3%

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

      2. distribute-lft-neg-out 77.3%

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

      3. distribute-frac-neg2 77.3%

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

      4. distribute-frac-neg 77.3%

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

      5. neg-mul-1 77.3%

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

      6. distribute-lft-neg-out 77.3%

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

      7. *-commutative 77.3%

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

      8. distribute-lft-neg-in 77.3%

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

      9. times-frac 77.3%

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

      10. metadata-eval 77.3%

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

      11. metadata-eval 77.3%

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

      12. associate--l+ 77.3%

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

      13. fma-define 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in x around 0 77.8%

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

      1. associate--l+ 77.8%

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

      2. div-sub 85.5%

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

   7. Simplified 85.5%

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

      1. pow2 85.5%

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

      2. pow2 85.5%

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

      3. difference-of-squares 91.4%

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

   9. Applied egg-rr 91.4%

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

   10. Taylor expanded in y around 0 72.2%

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

       1. associate-*r/ 77.2%

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

       2. +-commutative 77.2%

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

   12. Simplified 77.2%

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

   if 5.10000000000000014e70 < y < 4.4e191

   1. Initial program 52.9%

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

      1. remove-double-neg 52.9%

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

      2. distribute-lft-neg-out 52.9%

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

      3. distribute-frac-neg2 52.9%

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

      4. distribute-frac-neg 52.9%

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

      5. neg-mul-1 52.9%

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

      6. distribute-lft-neg-out 52.9%

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

      7. *-commutative 52.9%

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

      8. distribute-lft-neg-in 52.9%

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

      9. times-frac 52.9%

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

      10. metadata-eval 52.9%

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

      11. metadata-eval 52.9%

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

      12. associate--l+ 52.9%

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

      13. fma-define 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. associate--l+ 76.8%

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

      2. div-sub 76.8%

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

   7. Simplified 76.8%

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

      1. pow2 76.8%

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

      2. pow2 76.8%

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

      3. difference-of-squares 81.8%

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

   9. Applied egg-rr 81.8%

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

   10. Taylor expanded in x around 0 67.6%

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

   11. Taylor expanded in x around 0 67.6%

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

       1. neg-mul-1 67.6%

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

   13. Simplified 67.6%

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

   if 1.1e224 < y

   1. Initial program 5.9%

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

      1. remove-double-neg 5.9%

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

      2. distribute-lft-neg-out 5.9%

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

      3. distribute-frac-neg2 5.9%

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

      4. distribute-frac-neg 5.9%

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

      5. neg-mul-1 5.9%

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

      6. distribute-lft-neg-out 5.9%

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

      7. *-commutative 5.9%

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

      8. distribute-lft-neg-in 5.9%

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

      9. times-frac 5.9%

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

      10. metadata-eval 5.9%

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

      11. metadata-eval 5.9%

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

      12. associate--l+ 5.9%

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

      13. fma-define 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in y around inf 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+191}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z \cdot z}{y}\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+224}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{x\_m - z\_m}{y}\\ \mathbf{if}\;x\_m \cdot x\_m \leq 2 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \left(y + z\_m \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(x\_m + z\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (/ (- x_m z_m) y)))
   (if (<= (* x_m x_m) 2e+206)
     (* 0.5 (+ y (* z_m t_0)))
     (* 0.5 (* t_0 (+ x_m z_m))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double t_0 = (x_m - z_m) / y;
	double tmp;
	if ((x_m * x_m) <= 2e+206) {
		tmp = 0.5 * (y + (z_m * t_0));
	} else {
		tmp = 0.5 * (t_0 * (x_m + z_m));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_m - z_m) / y
    if ((x_m * x_m) <= 2d+206) then
        tmp = 0.5d0 * (y + (z_m * t_0))
    else
        tmp = 0.5d0 * (t_0 * (x_m + z_m))
    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 t_0 = (x_m - z_m) / y;
	double tmp;
	if ((x_m * x_m) <= 2e+206) {
		tmp = 0.5 * (y + (z_m * t_0));
	} else {
		tmp = 0.5 * (t_0 * (x_m + z_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	t_0 = (x_m - z_m) / y
	tmp = 0
	if (x_m * x_m) <= 2e+206:
		tmp = 0.5 * (y + (z_m * t_0))
	else:
		tmp = 0.5 * (t_0 * (x_m + z_m))
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(x_m - z_m) / y)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 2e+206)
		tmp = Float64(0.5 * Float64(y + Float64(z_m * t_0)));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(x_m + z_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (x_m - z_m) / y;
	tmp = 0.0;
	if ((x_m * x_m) <= 2e+206)
		tmp = 0.5 * (y + (z_m * t_0));
	else
		tmp = 0.5 * (t_0 * (x_m + z_m));
	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_] := Block[{t$95$0 = N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 2e+206], N[(0.5 * N[(y + N[(z$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(x$95$m + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 71.4%

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

      1. remove-double-neg 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-frac-neg2 71.4%

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

      4. distribute-frac-neg 71.4%

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

      5. neg-mul-1 71.4%

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

      6. distribute-lft-neg-out 71.4%

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

      7. *-commutative 71.4%

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

      8. distribute-lft-neg-in 71.4%

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

      9. times-frac 71.4%

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

      10. metadata-eval 71.4%

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

      11. metadata-eval 71.4%

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

      12. associate--l+ 71.4%

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

      13. fma-define 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in x around 0 88.4%

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

      1. associate--l+ 88.4%

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

      2. div-sub 90.2%

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

   7. Simplified 90.2%

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

      1. pow2 90.2%

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

      2. pow2 90.2%

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

      3. difference-of-squares 90.2%

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

   9. Applied egg-rr 90.2%

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

   10. Taylor expanded in x around 0 75.4%

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

   11. Taylor expanded in z around 0 83.8%

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

       1. +-commutative 83.8%

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

       2. neg-mul-1 83.8%

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

       3. sub-neg 83.8%

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

       4. div-sub 85.0%

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

   13. Simplified 85.0%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 70.9%

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

      1. remove-double-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. distribute-frac-neg2 70.9%

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

      4. distribute-frac-neg 70.9%

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

      5. neg-mul-1 70.9%

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

      6. distribute-lft-neg-out 70.9%

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

      7. *-commutative 70.9%

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

      8. distribute-lft-neg-in 70.9%

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

      9. times-frac 70.9%

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

      10. metadata-eval 70.9%

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

      11. metadata-eval 70.9%

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

      12. associate--l+ 70.9%

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

      13. fma-define 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. associate--l+ 58.3%

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

      2. div-sub 72.9%

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

   7. Simplified 72.9%

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

      1. pow2 72.9%

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

      2. pow2 72.9%

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

      3. difference-of-squares 87.6%

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

   9. Applied egg-rr 87.6%

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

   10. Taylor expanded in y around 0 84.1%

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

       1. associate-*r/ 91.5%

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

       2. +-commutative 91.5%

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

   12. Simplified 91.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \left(y + z \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.3 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x\_m \cdot \left(x\_m - z\_m\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + z\_m \cdot \frac{x\_m - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= z_m 3.3e-44)
   (* 0.5 (+ y (/ (* x_m (- x_m z_m)) y)))
   (* 0.5 (+ y (* 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 (z_m <= 3.3e-44) {
		tmp = 0.5 * (y + ((x_m * (x_m - z_m)) / y));
	} else {
		tmp = 0.5 * (y + (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 (z_m <= 3.3d-44) then
        tmp = 0.5d0 * (y + ((x_m * (x_m - z_m)) / y))
    else
        tmp = 0.5d0 * (y + (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 (z_m <= 3.3e-44) {
		tmp = 0.5 * (y + ((x_m * (x_m - z_m)) / y));
	} else {
		tmp = 0.5 * (y + (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 z_m <= 3.3e-44:
		tmp = 0.5 * (y + ((x_m * (x_m - z_m)) / y))
	else:
		tmp = 0.5 * (y + (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 (z_m <= 3.3e-44)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x_m * Float64(x_m - z_m)) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(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 (z_m <= 3.3e-44)
		tmp = 0.5 * (y + ((x_m * (x_m - z_m)) / y));
	else
		tmp = 0.5 * (y + (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[z$95$m, 3.3e-44], N[(0.5 * N[(y + N[(N[(x$95$m * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(z$95$m * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.30000000000000006e-44

   1. Initial program 73.0%

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

      1. remove-double-neg 73.0%

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

      2. distribute-lft-neg-out 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-frac-neg 73.0%

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

      5. neg-mul-1 73.0%

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

      6. distribute-lft-neg-out 73.0%

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

      7. *-commutative 73.0%

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

      8. distribute-lft-neg-in 73.0%

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

      9. times-frac 73.0%

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

      10. metadata-eval 73.0%

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

      11. metadata-eval 73.0%

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

      12. associate--l+ 73.0%

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

      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 81.9%

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

      1. associate--l+ 81.9%

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

      2. div-sub 86.5%

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

   7. Simplified 86.5%

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

      1. pow2 86.5%

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

      2. pow2 86.5%

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

      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. Taylor expanded in x around inf 77.4%

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

   if 3.30000000000000006e-44 < z

   1. Initial program 67.4%

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

      1. remove-double-neg 67.4%

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

      2. distribute-lft-neg-out 67.4%

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

      3. distribute-frac-neg2 67.4%

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

      4. distribute-frac-neg 67.4%

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

      5. neg-mul-1 67.4%

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

      6. distribute-lft-neg-out 67.4%

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

      7. *-commutative 67.4%

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

      8. distribute-lft-neg-in 67.4%

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

      9. times-frac 67.4%

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

      10. metadata-eval 67.4%

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

      11. metadata-eval 67.4%

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

      12. associate--l+ 67.4%

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

      13. fma-define 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. associate--l+ 67.0%

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

      2. div-sub 77.8%

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

   7. Simplified 77.8%

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

      1. pow2 77.8%

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

      2. pow2 77.8%

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

      3. difference-of-squares 85.1%

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

   9. Applied egg-rr 85.1%

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

   10. Taylor expanded in x around 0 72.7%

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

   11. Taylor expanded in z around 0 80.8%

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

       1. +-commutative 80.8%

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

       2. neg-mul-1 80.8%

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

       3. sub-neg 80.8%

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

       4. div-sub 83.2%

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

   13. Simplified 83.2%

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

Alternative 5: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 2 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z\_m \cdot z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{0.5 \cdot x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 2e+260)
   (* 0.5 (- y (/ (* z_m z_m) y)))
   (* x_m (/ (* 0.5 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) <= 2e+260) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else {
		tmp = x_m * ((0.5 * 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) <= 2d+260) then
        tmp = 0.5d0 * (y - ((z_m * z_m) / y))
    else
        tmp = x_m * ((0.5d0 * 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) <= 2e+260) {
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	} else {
		tmp = x_m * ((0.5 * 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) <= 2e+260:
		tmp = 0.5 * (y - ((z_m * z_m) / y))
	else:
		tmp = x_m * ((0.5 * 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) <= 2e+260)
		tmp = Float64(0.5 * Float64(y - Float64(Float64(z_m * z_m) / y)));
	else
		tmp = Float64(x_m * Float64(Float64(0.5 * 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) <= 2e+260)
		tmp = 0.5 * (y - ((z_m * z_m) / y));
	else
		tmp = x_m * ((0.5 * 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], 2e+260], N[(0.5 * N[(y - N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000013e260

   1. Initial program 73.1%

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

      1. remove-double-neg 73.1%

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

      2. distribute-lft-neg-out 73.1%

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

      3. distribute-frac-neg2 73.1%

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

      4. distribute-frac-neg 73.1%

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

      5. neg-mul-1 73.1%

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

      6. distribute-lft-neg-out 73.1%

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

      7. *-commutative 73.1%

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

      8. distribute-lft-neg-in 73.1%

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

      9. times-frac 73.1%

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

      10. metadata-eval 73.1%

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

      11. metadata-eval 73.1%

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

      12. associate--l+ 73.1%

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

      13. fma-define 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around 0 87.0%

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

      1. associate--l+ 87.0%

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

      2. div-sub 91.0%

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

   7. Simplified 91.0%

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

      1. pow2 91.0%

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

      2. pow2 91.0%

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

      3. difference-of-squares 91.0%

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

   9. Applied egg-rr 91.0%

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

   10. Taylor expanded in x around 0 74.0%

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

   11. Taylor expanded in x around 0 73.3%

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

       1. neg-mul-1 73.3%

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

   13. Simplified 73.3%

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

   if 2.00000000000000013e260 < (*.f64 x x)

   1. Initial program 67.1%

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

      1. remove-double-neg 67.1%

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

      2. distribute-lft-neg-out 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-frac-neg 67.1%

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

      5. neg-mul-1 67.1%

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

      6. distribute-lft-neg-out 67.1%

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

      7. *-commutative 67.1%

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

      8. distribute-lft-neg-in 67.1%

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

      9. times-frac 67.1%

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

      10. metadata-eval 67.1%

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

      11. metadata-eval 67.1%

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

      12. associate--l+ 67.1%

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

      13. fma-define 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in x around inf 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l/ 76.3%

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

      3. associate-*r/ 76.3%

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

   7. Simplified 76.3%

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

      1. associate-*r/ 76.3%

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

      2. clear-num 76.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

      3. *-commutative 76.3%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{0.5 \cdot {x}^{2}}}} \]

   9. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{0.5 \cdot {x}^{2}}}} \]
   10. Step-by-step derivation

       1. pow2 76.3%

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

       2. clear-num 76.3%

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

       3. *-commutative 76.3%

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

       4. associate-*r/ 76.3%

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

       5. associate-*l* 83.8%

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

   11. Applied egg-rr 83.8%

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

   12. Taylor expanded in x around 0 83.8%

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

       1. associate-*r/ 83.8%

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

       2. *-commutative 83.8%

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

   14. Simplified 83.8%

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

4. Final simplification 76.7%

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

Alternative 6: 55.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{x\_m - z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= y 1.05e+36) (* 0.5 (* 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 <= 1.05e+36) {
		tmp = 0.5 * (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 <= 1.05d+36) then
        tmp = 0.5d0 * (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 <= 1.05e+36) {
		tmp = 0.5 * (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 <= 1.05e+36:
		tmp = 0.5 * (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 <= 1.05e+36)
		tmp = Float64(0.5 * Float64(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 <= 1.05e+36)
		tmp = 0.5 * (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, 1.05e+36], N[(0.5 * N[(z$95$m * 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 < 1.05000000000000002e36

   1. Initial program 77.2%

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

      1. remove-double-neg 77.2%

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

      2. distribute-lft-neg-out 77.2%

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

      3. distribute-frac-neg2 77.2%

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

      4. distribute-frac-neg 77.2%

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

      5. neg-mul-1 77.2%

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

      6. distribute-lft-neg-out 77.2%

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

      7. *-commutative 77.2%

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

      8. distribute-lft-neg-in 77.2%

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

      9. times-frac 77.2%

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

      10. metadata-eval 77.2%

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

      11. metadata-eval 77.2%

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

      12. associate--l+ 77.2%

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

      13. fma-define 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 77.7%

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

      1. associate--l+ 77.7%

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

      2. div-sub 85.6%

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

   7. Simplified 85.6%

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

      1. pow2 85.6%

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

      2. pow2 85.6%

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

      3. difference-of-squares 91.7%

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

   9. Applied egg-rr 91.7%

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

   10. Taylor expanded in y around 0 72.5%

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

       1. associate-*r/ 77.1%

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

       2. +-commutative 77.1%

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

   12. Simplified 77.1%

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

   13. Taylor expanded in z around inf 45.1%

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

   if 1.05000000000000002e36 < y

   1. Initial program 38.8%

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

      1. remove-double-neg 38.8%

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

      2. distribute-lft-neg-out 38.8%

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

      3. distribute-frac-neg2 38.8%

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

      4. distribute-frac-neg 38.8%

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

      5. neg-mul-1 38.8%

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

      6. distribute-lft-neg-out 38.8%

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

      7. *-commutative 38.8%

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

      8. distribute-lft-neg-in 38.8%

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

      9. times-frac 38.8%

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

      10. metadata-eval 38.8%

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

      11. metadata-eval 38.8%

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

      12. associate--l+ 38.8%

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

      13. fma-define 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in y around inf 64.0%

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

Alternative 7: 52.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{+215}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{0.5 \cdot x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (* x_m x_m) 1e+215) (* 0.5 y) (* x_m (/ (* 0.5 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) <= 1e+215) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * ((0.5 * 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) <= 1d+215) then
        tmp = 0.5d0 * y
    else
        tmp = x_m * ((0.5d0 * 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) <= 1e+215) {
		tmp = 0.5 * y;
	} else {
		tmp = x_m * ((0.5 * 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) <= 1e+215:
		tmp = 0.5 * y
	else:
		tmp = x_m * ((0.5 * 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) <= 1e+215)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x_m * Float64(Float64(0.5 * 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) <= 1e+215)
		tmp = 0.5 * y;
	else
		tmp = x_m * ((0.5 * 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], 1e+215], N[(0.5 * y), $MachinePrecision], N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999907e214

   1. Initial program 71.6%

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

      1. remove-double-neg 71.6%

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

      2. distribute-lft-neg-out 71.6%

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

      3. distribute-frac-neg2 71.6%

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

      4. distribute-frac-neg 71.6%

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

      5. neg-mul-1 71.6%

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

      6. distribute-lft-neg-out 71.6%

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

      7. *-commutative 71.6%

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

      8. distribute-lft-neg-in 71.6%

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

      9. times-frac 71.6%

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

      10. metadata-eval 71.6%

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

      11. metadata-eval 71.6%

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

      12. associate--l+ 71.6%

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

      13. fma-define 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in y around inf 42.1%

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

   if 9.99999999999999907e214 < (*.f64 x x)

   1. Initial program 70.6%

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

      1. remove-double-neg 70.6%

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

      2. distribute-lft-neg-out 70.6%

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

      3. distribute-frac-neg2 70.6%

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

      4. distribute-frac-neg 70.6%

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

      5. neg-mul-1 70.6%

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

      6. distribute-lft-neg-out 70.6%

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

      7. *-commutative 70.6%

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

      8. distribute-lft-neg-in 70.6%

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

      9. times-frac 70.6%

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

      10. metadata-eval 70.6%

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

      11. metadata-eval 70.6%

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

      12. associate--l+ 70.6%

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

      13. fma-define 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in x around inf 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*l/ 72.3%

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

      3. associate-*r/ 72.3%

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

   7. Simplified 72.3%

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

      1. associate-*r/ 72.3%

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

      2. clear-num 72.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

      3. *-commutative 72.3%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{0.5 \cdot {x}^{2}}}} \]

   9. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{0.5 \cdot {x}^{2}}}} \]
   10. Step-by-step derivation

       1. pow2 72.3%

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

       2. clear-num 72.3%

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

       3. *-commutative 72.3%

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

       4. associate-*r/ 72.3%

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

       5. associate-*l* 78.8%

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

   11. Applied egg-rr 78.8%

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

   12. Taylor expanded in x around 0 78.8%

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

       1. associate-*r/ 78.8%

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

       2. *-commutative 78.8%

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

   14. Simplified 78.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+215}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= x_m 4.2e+107) (* 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 <= 4.2e+107) {
		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 <= 4.2d+107) 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 <= 4.2e+107) {
		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 <= 4.2e+107:
		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 (x_m <= 4.2e+107)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x_m * Float64(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 <= 4.2e+107)
		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[x$95$m, 4.2e+107], N[(0.5 * y), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1999999999999999e107

   1. Initial program 72.1%

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

      1. remove-double-neg 72.1%

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

      2. distribute-lft-neg-out 72.1%

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

      3. distribute-frac-neg2 72.1%

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

      4. distribute-frac-neg 72.1%

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

      5. neg-mul-1 72.1%

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

      6. distribute-lft-neg-out 72.1%

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

      7. *-commutative 72.1%

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

      8. distribute-lft-neg-in 72.1%

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

      9. times-frac 72.1%

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

      10. metadata-eval 72.1%

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

      11. metadata-eval 72.1%

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

      12. associate--l+ 72.1%

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

      13. fma-define 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in y around inf 34.5%

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

   if 4.1999999999999999e107 < x

   1. Initial program 67.2%

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

      1. remove-double-neg 67.2%

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

      2. distribute-lft-neg-out 67.2%

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

      3. distribute-frac-neg2 67.2%

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

      4. distribute-frac-neg 67.2%

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

      5. neg-mul-1 67.2%

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

      6. distribute-lft-neg-out 67.2%

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

      7. *-commutative 67.2%

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

      8. distribute-lft-neg-in 67.2%

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

      9. times-frac 67.2%

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

      10. metadata-eval 67.2%

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

      11. metadata-eval 67.2%

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

      12. associate--l+ 67.2%

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

      13. fma-define 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in x around inf 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*l/ 64.3%

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

      3. associate-*r/ 64.3%

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

   7. Simplified 64.3%

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

      1. associate-*r/ 64.3%

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

      2. clear-num 64.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

      3. *-commutative 64.3%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{0.5 \cdot {x}^{2}}}} \]

   9. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{0.5 \cdot {x}^{2}}}} \]
   10. Step-by-step derivation

       1. pow2 64.3%

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

       2. clear-num 64.3%

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

       3. *-commutative 64.3%

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

       4. associate-*r/ 64.3%

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

       5. associate-*l* 71.8%

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

   11. Applied egg-rr 71.8%

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

Alternative 9: 34.2% 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 71.2%

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

   1. remove-double-neg 71.2%

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

   2. distribute-lft-neg-out 71.2%

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

   3. distribute-frac-neg2 71.2%

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

   4. distribute-frac-neg 71.2%

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

   5. neg-mul-1 71.2%

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

   6. distribute-lft-neg-out 71.2%

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

   7. *-commutative 71.2%

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

   8. distribute-lft-neg-in 71.2%

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

   9. times-frac 71.2%

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

   10. metadata-eval 71.2%

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

   11. metadata-eval 71.2%

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

   12. associate--l+ 71.2%

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

   13. fma-define 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in y around inf 30.6%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024143 
(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)))