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

Percentage Accurate: 69.1% → 97.6%

Time: 10.3s

Alternatives: 7

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\left(z + x\right) \cdot \left(t\_0 \cdot \frac{1}{y\_m}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (- x z) y_m)))
   (*
    y_s
    (if (<= y_m 2.2e-78)
      (* 0.5 (* (+ z x) t_0))
      (* y_m (+ 0.5 (* 0.5 (* (+ z x) (* t_0 (/ 1.0 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 t_0 = (x - z) / y_m;
	double tmp;
	if (y_m <= 2.2e-78) {
		tmp = 0.5 * ((z + x) * t_0);
	} else {
		tmp = y_m * (0.5 + (0.5 * ((z + x) * (t_0 * (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (x - z) / y_m
    if (y_m <= 2.2d-78) then
        tmp = 0.5d0 * ((z + x) * t_0)
    else
        tmp = y_m * (0.5d0 + (0.5d0 * ((z + x) * (t_0 * (1.0d0 / 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 t_0 = (x - z) / y_m;
	double tmp;
	if (y_m <= 2.2e-78) {
		tmp = 0.5 * ((z + x) * t_0);
	} else {
		tmp = y_m * (0.5 + (0.5 * ((z + x) * (t_0 * (1.0 / 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):
	t_0 = (x - z) / y_m
	tmp = 0
	if y_m <= 2.2e-78:
		tmp = 0.5 * ((z + x) * t_0)
	else:
		tmp = y_m * (0.5 + (0.5 * ((z + x) * (t_0 * (1.0 / 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)
	t_0 = Float64(Float64(x - z) / y_m)
	tmp = 0.0
	if (y_m <= 2.2e-78)
		tmp = Float64(0.5 * Float64(Float64(z + x) * t_0));
	else
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(z + x) * Float64(t_0 * Float64(1.0 / 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)
	t_0 = (x - z) / y_m;
	tmp = 0.0;
	if (y_m <= 2.2e-78)
		tmp = 0.5 * ((z + x) * t_0);
	else
		tmp = y_m * (0.5 + (0.5 * ((z + x) * (t_0 * (1.0 / 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_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 2.2e-78], N[(0.5 * N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(t$95$0 * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1999999999999999e-78

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf 73.1%

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

      1. *-commutative 73.1%

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

   5. Simplified 73.1%

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

      1. unpow2 73.1%

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

      2. unpow2 73.1%

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

      3. difference-of-squares 78.5%

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

   7. Applied egg-rr 78.5%

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

   8. Taylor expanded in y around 0 69.7%

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

      1. associate-/l* 73.7%

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

      2. +-commutative 73.7%

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

   10. Simplified 73.7%

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

   if 2.1999999999999999e-78 < y

   1. Initial program 57.4%

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

   3. Taylor expanded in y around inf 71.9%

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

      1. *-commutative 71.9%

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

   5. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. difference-of-squares 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. associate-/l* 90.8%

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

   9. Applied egg-rr 90.8%

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

       1. +-commutative 90.8%

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

   11. Simplified 90.8%

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

       1. *-un-lft-identity 90.8%

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

       2. unpow2 90.8%

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

       3. times-frac 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(x \cdot \left(\frac{1}{y\_m} \cdot \frac{x}{y\_m}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* 0.5 (* (+ z x) (/ (- x z) y_m)))
      (if (<= t_0 4e+306)
        t_0
        (* y_m (+ 0.5 (* 0.5 (* x (* (/ 1.0 y_m) (/ 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 t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (t_0 <= 4e+306) {
		tmp = t_0;
	} else {
		tmp = y_m * (0.5 + (0.5 * (x * ((1.0 / y_m) * (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) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = 0.5d0 * ((z + x) * ((x - z) / y_m))
    else if (t_0 <= 4d+306) then
        tmp = t_0
    else
        tmp = y_m * (0.5d0 + (0.5d0 * (x * ((1.0d0 / y_m) * (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 t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (t_0 <= 4e+306) {
		tmp = t_0;
	} else {
		tmp = y_m * (0.5 + (0.5 * (x * ((1.0 / y_m) * (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):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.5 * ((z + x) * ((x - z) / y_m))
	elif t_0 <= 4e+306:
		tmp = t_0
	else:
		tmp = y_m * (0.5 + (0.5 * (x * ((1.0 / y_m) * (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)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (t_0 <= 4e+306)
		tmp = t_0;
	else
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(x * Float64(Float64(1.0 / y_m) * 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)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	elseif (t_0 <= 4e+306)
		tmp = t_0;
	else
		tmp = y_m * (0.5 + (0.5 * (x * ((1.0 / y_m) * (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_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], t$95$0, N[(y$95$m * N[(0.5 + N[(0.5 * N[(x * N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 0.0

   1. Initial program 72.8%

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

   3. Taylor expanded in y around inf 82.1%

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

      1. *-commutative 82.1%

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

   5. Simplified 82.1%

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

      1. unpow2 82.1%

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

      2. unpow2 82.1%

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

      3. difference-of-squares 82.1%

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

   7. Applied egg-rr 82.1%

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

   8. Taylor expanded in y around 0 55.2%

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

      1. associate-/l* 60.9%

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

      2. +-commutative 60.9%

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

   10. Simplified 60.9%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.00000000000000007e306

   1. Initial program 99.8%

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

   if 4.00000000000000007e306 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 52.6%

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

   3. Taylor expanded in y around inf 62.7%

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

      1. *-commutative 62.7%

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

   5. Simplified 62.7%

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

      1. unpow2 62.7%

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

      2. unpow2 62.7%

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

      3. difference-of-squares 75.5%

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

   7. Applied egg-rr 75.5%

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

   8. Taylor expanded in z around 0 24.0%

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

      1. distribute-lft-in 24.0%

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

      2. associate-+l+ 24.0%

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

      3. *-commutative 24.0%

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

      4. associate-*l* 24.0%

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

      5. *-commutative 24.0%

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

      6. associate-*r/ 24.0%

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

      7. associate-*l/ 27.9%

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

      8. associate-*l* 27.9%

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

      9. *-commutative 27.9%

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

      10. *-commutative 27.9%

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

      11. associate-*l/ 28.9%

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

      12. associate-*r/ 28.8%

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

      13. unpow2 28.8%

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

      14. associate-/l* 36.3%

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

      15. distribute-lft-in 38.3%

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

      16. +-commutative 38.3%

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

      17. distribute-lft-in 39.3%

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

   10. Simplified 59.9%

       \[\leadsto y \cdot \left(0.5 + \color{blue}{\left(x \cdot \left(\frac{x}{{y}^{2}} + 0\right)\right)} \cdot 0.5\right) \]
   11. Step-by-step derivation

       1. *-un-lft-identity 59.9%

          \[\leadsto y \cdot \left(0.5 + \left(x \cdot \left(\frac{\color{blue}{1 \cdot x}}{{y}^{2}} + 0\right)\right) \cdot 0.5\right) \]

       2. unpow2 59.9%

          \[\leadsto y \cdot \left(0.5 + \left(x \cdot \left(\frac{1 \cdot x}{\color{blue}{y \cdot y}} + 0\right)\right) \cdot 0.5\right) \]

       3. times-frac 67.1%

          \[\leadsto y \cdot \left(0.5 + \left(x \cdot \left(\color{blue}{\frac{1}{y} \cdot \frac{x}{y}} + 0\right)\right) \cdot 0.5\right) \]

   12. Applied egg-rr 67.1%

       \[\leadsto y \cdot \left(0.5 + \left(x \cdot \left(\color{blue}{\frac{1}{y} \cdot \frac{x}{y}} + 0\right)\right) \cdot 0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(x \cdot \left(\frac{1}{y} \cdot \frac{x}{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.5 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 6.5e-89)
    (* 0.5 (* (+ z x) (/ (- x z) y_m)))
    (if (<= y_m 3.1e+155)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* 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) {
	double tmp;
	if (y_m <= 6.5e-89) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 3.1e+155) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 6.5d-89) then
        tmp = 0.5d0 * ((z + x) * ((x - z) / y_m))
    else if (y_m <= 3.1d+155) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = y_m * 0.5d0
    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 <= 6.5e-89) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else if (y_m <= 3.1e+155) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 6.5e-89:
		tmp = 0.5 * ((z + x) * ((x - z) / y_m))
	elif y_m <= 3.1e+155:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = y_m * 0.5
	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 <= 6.5e-89)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	elseif (y_m <= 3.1e+155)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(y_m * 0.5);
	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 <= 6.5e-89)
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	elseif (y_m <= 3.1e+155)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = y_m * 0.5;
	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, 6.5e-89], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.1e+155], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 6.50000000000000034e-89

   1. Initial program 72.8%

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

   3. Taylor expanded in y around inf 72.5%

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

      1. *-commutative 72.5%

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

   5. Simplified 72.5%

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

      1. unpow2 72.5%

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

      2. unpow2 72.5%

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

      3. difference-of-squares 78.0%

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

   7. Applied egg-rr 78.0%

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

   8. Taylor expanded in y around 0 69.1%

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

      1. associate-/l* 73.1%

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

      2. +-commutative 73.1%

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

   10. Simplified 73.1%

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

   if 6.50000000000000034e-89 < y < 3.09999999999999989e155

   1. Initial program 91.4%

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

   if 3.09999999999999989e155 < y

   1. Initial program 9.3%

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

   3. Taylor expanded in y around inf 81.0%

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

      1. *-commutative 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y\_m}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+23}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{z}{-y\_m}\right)\\ \end{array} \end{array} \]

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 (<= z 9.2e-204)
    (* 0.5 (* (+ z x) (/ x y_m)))
    (if (<= z 1.05e+23) (* y_m 0.5) (* 0.5 (* (+ z x) (/ z (- 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 (z <= 9.2e-204) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} else if (z <= 1.05e+23) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * ((z + x) * (z / -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 (z <= 9.2d-204) then
        tmp = 0.5d0 * ((z + x) * (x / y_m))
    else if (z <= 1.05d+23) then
        tmp = y_m * 0.5d0
    else
        tmp = 0.5d0 * ((z + x) * (z / -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 (z <= 9.2e-204) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} else if (z <= 1.05e+23) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * ((z + x) * (z / -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 z <= 9.2e-204:
		tmp = 0.5 * ((z + x) * (x / y_m))
	elif z <= 1.05e+23:
		tmp = y_m * 0.5
	else:
		tmp = 0.5 * ((z + x) * (z / -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 (z <= 9.2e-204)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(x / y_m)));
	elseif (z <= 1.05e+23)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(z / Float64(-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 (z <= 9.2e-204)
		tmp = 0.5 * ((z + x) * (x / y_m));
	elseif (z <= 1.05e+23)
		tmp = y_m * 0.5;
	else
		tmp = 0.5 * ((z + x) * (z / -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[z, 9.2e-204], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+23], N[(y$95$m * 0.5), $MachinePrecision], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(z / (-y$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 9.1999999999999997e-204

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf 70.1%

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

      1. *-commutative 70.1%

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

   5. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. difference-of-squares 75.4%

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in y around 0 59.2%

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

      1. associate-/l* 62.7%

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

      2. +-commutative 62.7%

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

   10. Simplified 62.7%

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

   11. Taylor expanded in x around inf 40.7%

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

   if 9.1999999999999997e-204 < z < 1.0500000000000001e23

   1. Initial program 71.4%

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

   3. Taylor expanded in y around inf 51.6%

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

      1. *-commutative 51.6%

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

   5. Simplified 51.6%

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

   if 1.0500000000000001e23 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. difference-of-squares 80.7%

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

   7. Applied egg-rr 80.7%

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

   8. Taylor expanded in y around 0 81.0%

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

      1. associate-/l* 85.7%

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

      2. +-commutative 85.7%

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

   10. Simplified 85.7%

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

   11. Taylor expanded in x around 0 77.0%

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

       1. associate-*r/ 77.0%

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

       2. mul-1-neg 77.0%

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

   13. Simplified 77.0%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+23}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{z}{-y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.7 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 3.7e+122) (* 0.5 (* (+ z x) (/ (- x z) y_m))) (* 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) {
	double tmp;
	if (y_m <= 3.7e+122) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 3.7d+122) then
        tmp = 0.5d0 * ((z + x) * ((x - z) / y_m))
    else
        tmp = y_m * 0.5d0
    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 <= 3.7e+122) {
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 3.7e+122:
		tmp = 0.5 * ((z + x) * ((x - z) / y_m))
	else:
		tmp = y_m * 0.5
	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 <= 3.7e+122)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	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 <= 3.7e+122)
		tmp = 0.5 * ((z + x) * ((x - z) / y_m));
	else
		tmp = y_m * 0.5;
	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, 3.7e+122], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.6999999999999997e122

   1. Initial program 76.5%

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

   3. Taylor expanded in y around inf 76.2%

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

      1. *-commutative 76.2%

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

   5. Simplified 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

      3. difference-of-squares 81.6%

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

   7. Applied egg-rr 81.6%

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

   8. Taylor expanded in y around 0 68.3%

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

      1. associate-/l* 72.0%

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

      2. +-commutative 72.0%

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

   10. Simplified 72.0%

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

   if 3.6999999999999997e122 < y

   1. Initial program 19.7%

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

   3. Taylor expanded in y around inf 78.2%

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

      1. *-commutative 78.2%

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

   5. Simplified 78.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 2.8e-7) (* 0.5 (* (+ z x) (/ x y_m))) (* 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) {
	double tmp;
	if (y_m <= 2.8e-7) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 2.8d-7) then
        tmp = 0.5d0 * ((z + x) * (x / y_m))
    else
        tmp = y_m * 0.5d0
    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 <= 2.8e-7) {
		tmp = 0.5 * ((z + x) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 2.8e-7:
		tmp = 0.5 * ((z + x) * (x / y_m))
	else:
		tmp = y_m * 0.5
	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 <= 2.8e-7)
		tmp = Float64(0.5 * Float64(Float64(z + x) * Float64(x / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	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 <= 2.8e-7)
		tmp = 0.5 * ((z + x) * (x / y_m));
	else
		tmp = y_m * 0.5;
	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, 2.8e-7], N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.80000000000000019e-7

   1. Initial program 74.1%

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

   3. Taylor expanded in y around inf 73.8%

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

      1. *-commutative 73.8%

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

   5. Simplified 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

      3. difference-of-squares 79.9%

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

   7. Applied egg-rr 79.9%

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

   8. Taylor expanded in y around 0 70.4%

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

      1. associate-/l* 74.0%

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

      2. +-commutative 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in x around inf 41.1%

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

   if 2.80000000000000019e-7 < y

   1. Initial program 51.3%

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

   3. Taylor expanded in y around inf 65.3%

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

      1. *-commutative 65.3%

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

   5. Simplified 65.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\left(z + x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.1% 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 69.0%

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

3. Taylor expanded in y around inf 35.8%

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

   1. *-commutative 35.8%

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

5. Simplified 35.8%

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

6. Final simplification 35.8%

   \[\leadsto y \cdot 0.5 \]
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 2024059 
(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)))