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

Percentage Accurate: 68.8% → 96.0%

Time: 10.4s

Alternatives: 5

Speedup: 0.7×

• 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: 68.8% 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: 96.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m - z \cdot z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m - z \cdot \frac{z}{y\_m}\right) + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 10.0)
    (* 0.5 (/ (fma x x (- (* y_m y_m) (* z z))) y_m))
    (* 0.5 (+ (- y_m (* z (/ z y_m))) (* x (/ x y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 10.0) {
		tmp = 0.5 * (fma(x, x, ((y_m * y_m) - (z * z))) / y_m);
	} else {
		tmp = 0.5 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 10.0)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y_m * y_m) - Float64(z * z))) / y_m));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m - Float64(z * Float64(z / y_m))) + Float64(x * Float64(x / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 10.0], N[(0.5 * N[(N[(x * x + N[(N[(y$95$m * y$95$m), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 10

   1. Initial program 81.6%

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

      1. remove-double-neg 81.6%

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

      2. distribute-lft-neg-out 81.6%

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

      3. distribute-frac-neg2 81.6%

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

      4. distribute-frac-neg 81.6%

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

      5. neg-mul-1 81.6%

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

      6. distribute-lft-neg-out 81.6%

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

      7. *-commutative 81.6%

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

      8. distribute-lft-neg-in 81.6%

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

      9. times-frac 81.6%

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

      10. metadata-eval 81.6%

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

      11. metadata-eval 81.6%

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

      12. associate--l+ 81.6%

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

      13. fma-define 85.9%

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

   3. Simplified 85.9%

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

   if 10 < y

   1. Initial program 49.3%

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

      1. remove-double-neg 49.3%

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

      2. distribute-lft-neg-out 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. neg-mul-1 49.3%

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

      6. distribute-lft-neg-out 49.3%

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

      7. *-commutative 49.3%

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

      8. distribute-lft-neg-in 49.3%

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

      9. times-frac 49.3%

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

      10. metadata-eval 49.3%

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

      11. metadata-eval 49.3%

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

      12. associate--l+ 49.3%

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

      13. fma-define 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in z around inf 47.8%

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

      1. associate--l+ 47.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 47.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. associate-/l* 49.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{x \cdot \frac{x}{y \cdot {z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 49.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \color{blue}{\mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, -\frac{1}{y}\right)}\right)\right) \]

      5. distribute-neg-frac 49.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \color{blue}{\frac{-1}{y}}\right)\right)\right) \]

      6. metadata-eval 49.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{\color{blue}{-1}}{y}\right)\right)\right) \]

   7. Simplified 49.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around 0 76.0%

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

      1. associate-+r+ 76.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + -1 \cdot \frac{{z}^{2}}{y}\right) + \frac{{x}^{2}}{y}\right)} \]

      2. mul-1-neg 76.0%

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

      3. sub-neg 76.0%

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

   10. Simplified 76.0%

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

       1. unpow2 76.0%

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

       2. *-un-lft-identity 76.0%

          \[\leadsto 0.5 \cdot \left(\left(y - \frac{z \cdot z}{\color{blue}{1 \cdot y}}\right) + \frac{{x}^{2}}{y}\right) \]

       3. times-frac 81.7%

          \[\leadsto 0.5 \cdot \left(\left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) + \frac{{x}^{2}}{y}\right) \]

   12. Applied egg-rr 81.7%

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

       1. unpow2 81.7%

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

       2. *-un-lft-identity 81.7%

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

       3. times-frac 98.5%

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

   14. Applied egg-rr 98.5%

       \[\leadsto 0.5 \cdot \left(\left(y - \frac{z}{1} \cdot \frac{z}{y}\right) + \color{blue}{\frac{x}{1} \cdot \frac{x}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

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

Alternative 2: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + \frac{x}{\frac{y\_m}{x}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* y_m 2.0))))
   (* y_s (if (<= t_0 -2e-57) t_0 (* 0.5 (+ y_m (/ x (/ y_m x))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e-57) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y_m + (x / (y_m / x)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-2d-57)) then
        tmp = t_0
    else
        tmp = 0.5d0 * (y_m + (x / (y_m / x)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e-57) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y_m + (x / (y_m / x)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -2e-57:
		tmp = t_0
	else:
		tmp = 0.5 * (y_m + (x / (y_m / x)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-57)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(x / Float64(y_m / x))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-57)
		tmp = t_0;
	else
		tmp = 0.5 * (y_m + (x / (y_m / x)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-57], t$95$0, N[(0.5 * N[(y$95$m + N[(x / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999991e-57

   1. Initial program 87.5%

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

   if -1.99999999999999991e-57 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.1%

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

      1. remove-double-neg 62.1%

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

      2. distribute-lft-neg-out 62.1%

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

      3. distribute-frac-neg2 62.1%

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

      4. distribute-frac-neg 62.1%

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

      5. neg-mul-1 62.1%

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

      6. distribute-lft-neg-out 62.1%

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

      7. *-commutative 62.1%

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

      8. distribute-lft-neg-in 62.1%

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

      9. times-frac 62.1%

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

      10. metadata-eval 62.1%

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

      11. metadata-eval 62.1%

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

      12. associate--l+ 62.1%

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

      13. fma-define 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in z around inf 51.4%

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

      1. associate--l+ 51.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 51.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. associate-/l* 57.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{x \cdot \frac{x}{y \cdot {z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 57.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \color{blue}{\mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, -\frac{1}{y}\right)}\right)\right) \]

      5. distribute-neg-frac 57.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \color{blue}{\frac{-1}{y}}\right)\right)\right) \]

      6. metadata-eval 57.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{\color{blue}{-1}}{y}\right)\right)\right) \]

   7. Simplified 57.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around 0 74.3%

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

      1. associate-+r+ 74.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + -1 \cdot \frac{{z}^{2}}{y}\right) + \frac{{x}^{2}}{y}\right)} \]

      2. mul-1-neg 74.3%

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

      3. sub-neg 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in z around 0 54.9%

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

       1. +-commutative 54.9%

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

       2. unpow2 54.9%

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

       3. associate-*r/ 64.7%

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

       4. fma-define 64.7%

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

   13. Simplified 64.7%

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

       1. fma-undefine 64.7%

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

       2. clear-num 64.7%

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

       3. un-div-inv 64.7%

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

   15. Applied egg-rr 64.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m - z \cdot \frac{z}{y\_m}\right) + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5.5e-120)
    (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* y_m 2.0))
    (* 0.5 (+ (- y_m (* z (/ z y_m))) (* x (/ x y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.5e-120) {
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 5.5d-120) then
        tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.5e-120) {
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 5.5e-120:
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5.5e-120)
		tmp = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m - Float64(z * Float64(z / y_m))) + Float64(x * Float64(x / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5.5e-120)
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * ((y_m - (z * (z / y_m))) + (x * (x / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5.5e-120], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000001e-120

   1. Initial program 78.6%

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

   if 5.5000000000000001e-120 < y

   1. Initial program 62.9%

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

      1. remove-double-neg 62.9%

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

      2. distribute-lft-neg-out 62.9%

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

      3. distribute-frac-neg2 62.9%

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

      4. distribute-frac-neg 62.9%

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

      5. neg-mul-1 62.9%

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

      6. distribute-lft-neg-out 62.9%

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

      7. *-commutative 62.9%

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

      8. distribute-lft-neg-in 62.9%

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

      9. times-frac 62.9%

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

      10. metadata-eval 62.9%

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

      11. metadata-eval 62.9%

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

      12. associate--l+ 62.9%

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

      13. fma-define 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in z around inf 58.3%

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

      1. associate--l+ 58.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 58.3%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. associate-/l* 59.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{x \cdot \frac{x}{y \cdot {z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 59.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \color{blue}{\mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, -\frac{1}{y}\right)}\right)\right) \]

      5. distribute-neg-frac 59.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \color{blue}{\frac{-1}{y}}\right)\right)\right) \]

      6. metadata-eval 59.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{\color{blue}{-1}}{y}\right)\right)\right) \]

   7. Simplified 59.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around 0 82.4%

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

      1. associate-+r+ 82.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + -1 \cdot \frac{{z}^{2}}{y}\right) + \frac{{x}^{2}}{y}\right)} \]

      2. mul-1-neg 82.4%

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

      3. sub-neg 82.4%

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

   10. Simplified 82.4%

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

       1. unpow2 82.4%

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

       2. *-un-lft-identity 82.4%

          \[\leadsto 0.5 \cdot \left(\left(y - \frac{z \cdot z}{\color{blue}{1 \cdot y}}\right) + \frac{{x}^{2}}{y}\right) \]

       3. times-frac 86.6%

          \[\leadsto 0.5 \cdot \left(\left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) + \frac{{x}^{2}}{y}\right) \]

   12. Applied egg-rr 86.6%

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

       1. unpow2 86.6%

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

       2. *-un-lft-identity 86.6%

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

       3. times-frac 98.9%

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

   14. Applied egg-rr 98.9%

       \[\leadsto 0.5 \cdot \left(\left(y - \frac{z}{1} \cdot \frac{z}{y}\right) + \color{blue}{\frac{x}{1} \cdot \frac{x}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y - z \cdot \frac{z}{y}\right) + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot \left(y\_m + \frac{x}{\frac{y\_m}{x}}\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (* 0.5 (+ y_m (/ x (/ y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * (y_m + (x / (y_m / x))));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (0.5d0 * (y_m + (x / (y_m / x))))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * (y_m + (x / (y_m / x))));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (0.5 * (y_m + (x / (y_m / x))))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * Float64(y_m + Float64(x / Float64(y_m / x)))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (0.5 * (y_m + (x / (y_m / x))));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(0.5 * N[(y$95$m + N[(x / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

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

   1. remove-double-neg 72.6%

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

   2. distribute-lft-neg-out 72.6%

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

   3. distribute-frac-neg2 72.6%

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

   4. distribute-frac-neg 72.6%

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

   5. neg-mul-1 72.6%

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

   6. distribute-lft-neg-out 72.6%

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

   7. *-commutative 72.6%

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

   8. distribute-lft-neg-in 72.6%

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

   9. times-frac 72.6%

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

   10. metadata-eval 72.6%

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

   11. metadata-eval 72.6%

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

   12. associate--l+ 72.6%

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

   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 z around inf 61.1%

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

   1. associate--l+ 61.1%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

   2. unpow2 61.1%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

   3. associate-/l* 64.7%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{x \cdot \frac{x}{y \cdot {z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

   4. fmm-def 64.7%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \color{blue}{\mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, -\frac{1}{y}\right)}\right)\right) \]

   5. distribute-neg-frac 64.7%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \color{blue}{\frac{-1}{y}}\right)\right)\right) \]

   6. metadata-eval 64.7%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{\color{blue}{-1}}{y}\right)\right)\right) \]

7. Simplified 64.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(x, \frac{x}{y \cdot {z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]

8. Taylor expanded in z around 0 81.4%

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

   1. associate-+r+ 81.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + -1 \cdot \frac{{z}^{2}}{y}\right) + \frac{{x}^{2}}{y}\right)} \]

   2. mul-1-neg 81.4%

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

   3. sub-neg 81.4%

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

10. Simplified 81.4%

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

11. Taylor expanded in z around 0 57.8%

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

    1. +-commutative 57.8%

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

    2. unpow2 57.8%

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

    3. associate-*r/ 63.5%

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

    4. fma-define 63.5%

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

13. Simplified 63.5%

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

    1. fma-undefine 63.5%

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

    2. clear-num 63.5%

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

    3. un-div-inv 63.5%

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

15. Applied egg-rr 63.5%

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

16. Final simplification 63.5%

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

Alternative 5: 34.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m * 0.5d0)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m * 0.5)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * 0.5))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m * 0.5);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in y around inf 34.2%

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

   1. *-commutative 34.2%

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

5. Simplified 34.2%

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

6. Final simplification 34.2%

   \[\leadsto y \cdot 0.5 \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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