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

Percentage Accurate: 69.2% → 97.0%

Time: 10.3s

Alternatives: 10

Speedup: 0.9×

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{hypot}\left(x, y\_m\right), \frac{\mathsf{hypot}\left(x, y\_m\right)}{y\_m \cdot 2}, \left(z \cdot \frac{0.5}{y\_m}\right) \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

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 5e-187)
    (* (- x z) (* (+ x z) (/ 0.5 y_m)))
    (fma
     (hypot x y_m)
     (/ (hypot x y_m) (* y_m 2.0))
     (* (* z (/ 0.5 y_m)) (- z))))))

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 <= 5e-187) {
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	} else {
		tmp = fma(hypot(x, y_m), (hypot(x, y_m) / (y_m * 2.0)), ((z * (0.5 / y_m)) * -z));
	}
	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 <= 5e-187)
		tmp = Float64(Float64(x - z) * Float64(Float64(x + z) * Float64(0.5 / y_m)));
	else
		tmp = fma(hypot(x, y_m), Float64(hypot(x, y_m) / Float64(y_m * 2.0)), Float64(Float64(z * Float64(0.5 / y_m)) * Float64(-z)));
	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, 5e-187], N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision] * (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999996e-187

   1. Initial program 74.6%

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

   3. Taylor expanded in y around 0 63.2%

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

      1. associate-*r/ 63.2%

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

      2. *-commutative 63.2%

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

      3. associate-/l* 63.2%

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

   5. Simplified 63.2%

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

      1. unpow2 63.2%

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

      2. unpow2 63.2%

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

      3. difference-of-squares 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 73.1%

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

   9. Applied egg-rr 73.1%

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

   if 4.9999999999999996e-187 < y

   1. Initial program 69.0%

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

      1. div-sub 68.0%

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

      2. add-sqr-sqrt 68.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}}{y \cdot 2} - \frac{z \cdot z}{y \cdot 2} \]

      3. associate-/l* 68.1%

         \[\leadsto \color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \frac{\sqrt{x \cdot x + y \cdot y}}{y \cdot 2}} - \frac{z \cdot z}{y \cdot 2} \]

      4. fma-neg 68.1%

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

      5. hypot-define 68.1%

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

      6. hypot-define 88.8%

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

      7. div-inv 88.8%

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

      8. pow2 88.8%

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

      9. *-commutative 88.8%

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

      10. associate-/r* 88.8%

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

      11. metadata-eval 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. *-commutative 88.8%

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

      2. unpow2 88.8%

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

      3. associate-*r* 95.7%

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

   6. Applied egg-rr 95.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{hypot}\left(x, y\right), \frac{\mathsf{hypot}\left(x, y\right)}{y \cdot 2}, \left(z \cdot \frac{0.5}{y}\right) \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.6 \cdot 10^{-195}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + z \cdot \left(z \cdot \frac{-1}{y\_m}\right)\right)\\ \end{array} \end{array} \]

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 7.6e-195)
    (* (- x z) (* (+ x z) (/ 0.5 y_m)))
    (if (<= y_m 1.25e+18)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* 0.5 (+ y_m (* z (* z (/ -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 tmp;
	if (y_m <= 7.6e-195) {
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	} else if (y_m <= 1.25e+18) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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) :: tmp
    if (y_m <= 7.6d-195) then
        tmp = (x - z) * ((x + z) * (0.5d0 / y_m))
    else if (y_m <= 1.25d+18) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m + (z * (z * ((-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 tmp;
	if (y_m <= 7.6e-195) {
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	} else if (y_m <= 1.25e+18) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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):
	tmp = 0
	if y_m <= 7.6e-195:
		tmp = (x - z) * ((x + z) * (0.5 / y_m))
	elif y_m <= 1.25e+18:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m + (z * (z * (-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)
	tmp = 0.0
	if (y_m <= 7.6e-195)
		tmp = Float64(Float64(x - z) * Float64(Float64(x + z) * Float64(0.5 / y_m)));
	elseif (y_m <= 1.25e+18)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(z * Float64(z * 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)
	tmp = 0.0;
	if (y_m <= 7.6e-195)
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	elseif (y_m <= 1.25e+18)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m + (z * (z * (-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_] := N[(y$95$s * If[LessEqual[y$95$m, 7.6e-195], N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.25e+18], 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[(0.5 * N[(y$95$m + N[(z * N[(z * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 7.60000000000000026e-195

   1. Initial program 74.3%

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

   3. Taylor expanded in y around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-/l* 62.7%

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

   5. Simplified 62.7%

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

      1. unpow2 62.7%

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

      2. unpow2 62.7%

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

      3. difference-of-squares 69.8%

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

   7. Applied egg-rr 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*l* 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 7.60000000000000026e-195 < y < 1.25e18

   1. Initial program 93.3%

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

   if 1.25e18 < y

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. div-sub 43.9%

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

      2. unpow2 43.9%

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

      3. associate-/l* 72.7%

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

      4. *-inverses 72.7%

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

      5. *-rgt-identity 72.7%

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

   5. Simplified 72.7%

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

      1. div-inv 72.7%

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

      2. unpow2 72.7%

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

      3. associate-*l* 82.0%

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

   7. Applied egg-rr 82.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-195}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z}{\frac{y\_m}{z}}\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x} \cdot \frac{y\_m}{x}}\\ \end{array} \end{array} \]

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 (<= x 7e+132)
    (* 0.5 (- y_m (/ z (/ y_m z))))
    (if (<= x 2.4e+171)
      (/ x (* y_m (/ 2.0 x)))
      (if (<= x 2.7e+188)
        (* 0.5 (- y_m (* z (/ z y_m))))
        (/ 1.0 (* (/ 2.0 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 tmp;
	if (x <= 7e+132) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else if (x <= 2.4e+171) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 2.7e+188) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 1.0 / ((2.0 / 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) :: tmp
    if (x <= 7d+132) then
        tmp = 0.5d0 * (y_m - (z / (y_m / z)))
    else if (x <= 2.4d+171) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (x <= 2.7d+188) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = 1.0d0 / ((2.0d0 / 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 tmp;
	if (x <= 7e+132) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else if (x <= 2.4e+171) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 2.7e+188) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 1.0 / ((2.0 / 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):
	tmp = 0
	if x <= 7e+132:
		tmp = 0.5 * (y_m - (z / (y_m / z)))
	elif x <= 2.4e+171:
		tmp = x / (y_m * (2.0 / x))
	elif x <= 2.7e+188:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = 1.0 / ((2.0 / 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)
	tmp = 0.0
	if (x <= 7e+132)
		tmp = Float64(0.5 * Float64(y_m - Float64(z / Float64(y_m / z))));
	elseif (x <= 2.4e+171)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (x <= 2.7e+188)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 / 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)
	tmp = 0.0;
	if (x <= 7e+132)
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	elseif (x <= 2.4e+171)
		tmp = x / (y_m * (2.0 / x));
	elseif (x <= 2.7e+188)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = 1.0 / ((2.0 / 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_] := N[(y$95$s * If[LessEqual[x, 7e+132], N[(0.5 * N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e+171], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+188], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(2.0 / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if x < 7.00000000000000041e132

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0 53.9%

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

      1. div-sub 53.9%

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

      2. unpow2 53.9%

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

      3. associate-/l* 69.7%

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

      4. *-inverses 69.7%

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

      5. *-rgt-identity 69.7%

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

   5. Simplified 69.7%

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

      1. unpow2 69.7%

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

      2. associate-/l* 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. clear-num 74.5%

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

      2. un-div-inv 74.6%

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

   9. Applied egg-rr 74.6%

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

   if 7.00000000000000041e132 < x < 2.39999999999999998e171

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

      2. times-frac 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.4%

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

      3. frac-times 96.6%

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

      4. *-un-lft-identity 96.6%

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

   7. Applied egg-rr 96.6%

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

   if 2.39999999999999998e171 < x < 2.7e188

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. div-sub 68.1%

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

      2. unpow2 68.1%

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

      3. associate-/l* 68.1%

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

      4. *-inverses 68.1%

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

      5. *-rgt-identity 68.1%

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

   5. Simplified 68.1%

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

      1. unpow2 68.1%

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

      2. associate-/l* 68.1%

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

   7. Applied egg-rr 68.1%

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

   if 2.7e188 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.1%

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

   5. Applied egg-rr 70.1%

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

      1. clear-num 70.1%

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

      2. clear-num 70.1%

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

      3. frac-times 70.1%

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

      4. metadata-eval 70.1%

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

   7. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot \frac{2}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x} \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z}{\frac{y\_m}{z}}\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+170}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \end{array} \end{array} \]

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 (<= x 4e+133)
    (* 0.5 (- y_m (/ z (/ y_m z))))
    (if (<= x 4.4e+170)
      (/ x (* y_m (/ 2.0 x)))
      (if (<= x 2.8e+188)
        (* 0.5 (- y_m (* z (/ z y_m))))
        (* x (* x (/ 0.5 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 (x <= 4e+133) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else if (x <= 4.4e+170) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 2.8e+188) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = x * (x * (0.5 / 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 (x <= 4d+133) then
        tmp = 0.5d0 * (y_m - (z / (y_m / z)))
    else if (x <= 4.4d+170) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (x <= 2.8d+188) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = x * (x * (0.5d0 / 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 (x <= 4e+133) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else if (x <= 4.4e+170) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 2.8e+188) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = x * (x * (0.5 / 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 x <= 4e+133:
		tmp = 0.5 * (y_m - (z / (y_m / z)))
	elif x <= 4.4e+170:
		tmp = x / (y_m * (2.0 / x))
	elif x <= 2.8e+188:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = x * (x * (0.5 / 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 (x <= 4e+133)
		tmp = Float64(0.5 * Float64(y_m - Float64(z / Float64(y_m / z))));
	elseif (x <= 4.4e+170)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (x <= 2.8e+188)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / 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 (x <= 4e+133)
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	elseif (x <= 4.4e+170)
		tmp = x / (y_m * (2.0 / x));
	elseif (x <= 2.8e+188)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = x * (x * (0.5 / 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[x, 4e+133], N[(0.5 * N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+170], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+188], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if x < 4.0000000000000001e133

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0 53.9%

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

      1. div-sub 53.9%

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

      2. unpow2 53.9%

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

      3. associate-/l* 69.7%

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

      4. *-inverses 69.7%

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

      5. *-rgt-identity 69.7%

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

   5. Simplified 69.7%

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

      1. unpow2 69.7%

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

      2. associate-/l* 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. clear-num 74.5%

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

      2. un-div-inv 74.6%

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

   9. Applied egg-rr 74.6%

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

   if 4.0000000000000001e133 < x < 4.39999999999999978e170

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

      2. times-frac 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.4%

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

      3. frac-times 96.6%

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

      4. *-un-lft-identity 96.6%

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

   7. Applied egg-rr 96.6%

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

   if 4.39999999999999978e170 < x < 2.7999999999999998e188

   1. Initial program 2.7%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. div-sub 68.1%

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

      2. unpow2 68.1%

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

      3. associate-/l* 68.1%

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

      4. *-inverses 68.1%

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

      5. *-rgt-identity 68.1%

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

   5. Simplified 68.1%

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

      1. unpow2 68.1%

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

      2. associate-/l* 68.1%

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

   7. Applied egg-rr 68.1%

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

   if 2.7999999999999998e188 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf 66.3%

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

      1. div-inv 66.3%

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

      2. unpow2 66.3%

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

      3. associate-*l* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-/r* 70.1%

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

      6. metadata-eval 70.1%

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

   5. Applied egg-rr 70.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+170}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+188}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (- y_m (* z (/ z y_m))))))
   (*
    y_s
    (if (<= x 3.8e+132)
      t_0
      (if (<= x 7e+169)
        (/ x (* y_m (/ 2.0 x)))
        (if (<= x 4.6e+188) t_0 (* x (* x (/ 0.5 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 = 0.5 * (y_m - (z * (z / y_m)));
	double tmp;
	if (x <= 3.8e+132) {
		tmp = t_0;
	} else if (x <= 7e+169) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 4.6e+188) {
		tmp = t_0;
	} else {
		tmp = x * (x * (0.5 / 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 = 0.5d0 * (y_m - (z * (z / y_m)))
    if (x <= 3.8d+132) then
        tmp = t_0
    else if (x <= 7d+169) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (x <= 4.6d+188) then
        tmp = t_0
    else
        tmp = x * (x * (0.5d0 / 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 = 0.5 * (y_m - (z * (z / y_m)));
	double tmp;
	if (x <= 3.8e+132) {
		tmp = t_0;
	} else if (x <= 7e+169) {
		tmp = x / (y_m * (2.0 / x));
	} else if (x <= 4.6e+188) {
		tmp = t_0;
	} else {
		tmp = x * (x * (0.5 / 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 = 0.5 * (y_m - (z * (z / y_m)))
	tmp = 0
	if x <= 3.8e+132:
		tmp = t_0
	elif x <= 7e+169:
		tmp = x / (y_m * (2.0 / x))
	elif x <= 4.6e+188:
		tmp = t_0
	else:
		tmp = x * (x * (0.5 / 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(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))))
	tmp = 0.0
	if (x <= 3.8e+132)
		tmp = t_0;
	elseif (x <= 7e+169)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (x <= 4.6e+188)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / 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 = 0.5 * (y_m - (z * (z / y_m)));
	tmp = 0.0;
	if (x <= 3.8e+132)
		tmp = t_0;
	elseif (x <= 7e+169)
		tmp = x / (y_m * (2.0 / x));
	elseif (x <= 4.6e+188)
		tmp = t_0;
	else
		tmp = x * (x * (0.5 / 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[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 3.8e+132], t$95$0, If[LessEqual[x, 7e+169], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e+188], t$95$0, N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 3.80000000000000006e132 or 7.00000000000000038e169 < x < 4.60000000000000023e188

   1. Initial program 74.2%

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

   3. Taylor expanded in x around 0 54.1%

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

      1. div-sub 54.1%

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

      2. unpow2 54.1%

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

      3. associate-/l* 69.7%

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

      4. *-inverses 69.7%

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

      5. *-rgt-identity 69.7%

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

   5. Simplified 69.7%

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

      1. unpow2 69.7%

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

      2. associate-/l* 74.5%

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

   7. Applied egg-rr 74.5%

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

   if 3.80000000000000006e132 < x < 7.00000000000000038e169

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

      2. times-frac 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. clear-num 96.4%

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

      3. frac-times 96.6%

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

      4. *-un-lft-identity 96.6%

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

   7. Applied egg-rr 96.6%

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

   if 4.60000000000000023e188 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf 66.3%

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

      1. div-inv 66.3%

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

      2. unpow2 66.3%

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

      3. associate-*l* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-/r* 70.1%

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

      6. metadata-eval 70.1%

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

   5. Applied egg-rr 70.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+188}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.7 \cdot 10^{+17}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + z \cdot \left(z \cdot \frac{-1}{y\_m}\right)\right)\\ \end{array} \end{array} \]

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 5.7e+17)
    (* (- x z) (* (+ x z) (/ 0.5 y_m)))
    (* 0.5 (+ y_m (* z (* z (/ -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 tmp;
	if (y_m <= 5.7e+17) {
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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) :: tmp
    if (y_m <= 5.7d+17) then
        tmp = (x - z) * ((x + z) * (0.5d0 / y_m))
    else
        tmp = 0.5d0 * (y_m + (z * (z * ((-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 tmp;
	if (y_m <= 5.7e+17) {
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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):
	tmp = 0
	if y_m <= 5.7e+17:
		tmp = (x - z) * ((x + z) * (0.5 / y_m))
	else:
		tmp = 0.5 * (y_m + (z * (z * (-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)
	tmp = 0.0
	if (y_m <= 5.7e+17)
		tmp = Float64(Float64(x - z) * Float64(Float64(x + z) * Float64(0.5 / y_m)));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(z * Float64(z * 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)
	tmp = 0.0;
	if (y_m <= 5.7e+17)
		tmp = (x - z) * ((x + z) * (0.5 / y_m));
	else
		tmp = 0.5 * (y_m + (z * (z * (-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_] := N[(y$95$s * If[LessEqual[y$95$m, 5.7e+17], N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(z * N[(z * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.7e17

   1. Initial program 78.6%

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

   3. Taylor expanded in y around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-/l* 66.8%

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

   5. Simplified 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. difference-of-squares 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. associate-*l* 76.0%

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

   9. Applied egg-rr 76.0%

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

   if 5.7e17 < y

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. div-sub 43.9%

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

      2. unpow2 43.9%

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

      3. associate-/l* 72.7%

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

      4. *-inverses 72.7%

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

      5. *-rgt-identity 72.7%

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

   5. Simplified 72.7%

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

      1. div-inv 72.7%

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

      2. unpow2 72.7%

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

      3. associate-*l* 82.0%

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

   7. Applied egg-rr 82.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{+17}:\\ \;\;\;\;\left(x - z\right) \cdot \left(\left(x + z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.14 \cdot 10^{+18}:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(\left(x - z\right) \cdot \left(x + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + z \cdot \left(z \cdot \frac{-1}{y\_m}\right)\right)\\ \end{array} \end{array} \]

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 1.14e+18)
    (* (/ 0.5 y_m) (* (- x z) (+ x z)))
    (* 0.5 (+ y_m (* z (* z (/ -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 tmp;
	if (y_m <= 1.14e+18) {
		tmp = (0.5 / y_m) * ((x - z) * (x + z));
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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) :: tmp
    if (y_m <= 1.14d+18) then
        tmp = (0.5d0 / y_m) * ((x - z) * (x + z))
    else
        tmp = 0.5d0 * (y_m + (z * (z * ((-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 tmp;
	if (y_m <= 1.14e+18) {
		tmp = (0.5 / y_m) * ((x - z) * (x + z));
	} else {
		tmp = 0.5 * (y_m + (z * (z * (-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):
	tmp = 0
	if y_m <= 1.14e+18:
		tmp = (0.5 / y_m) * ((x - z) * (x + z))
	else:
		tmp = 0.5 * (y_m + (z * (z * (-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)
	tmp = 0.0
	if (y_m <= 1.14e+18)
		tmp = Float64(Float64(0.5 / y_m) * Float64(Float64(x - z) * Float64(x + z)));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(z * Float64(z * 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)
	tmp = 0.0;
	if (y_m <= 1.14e+18)
		tmp = (0.5 / y_m) * ((x - z) * (x + z));
	else
		tmp = 0.5 * (y_m + (z * (z * (-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_] := N[(y$95$s * If[LessEqual[y$95$m, 1.14e+18], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(z * N[(z * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.14e18

   1. Initial program 78.6%

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

   3. Taylor expanded in y around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-/l* 66.8%

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

   5. Simplified 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. difference-of-squares 73.7%

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

   7. Applied egg-rr 73.7%

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

   if 1.14e18 < y

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. div-sub 43.9%

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

      2. unpow2 43.9%

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

      3. associate-/l* 72.7%

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

      4. *-inverses 72.7%

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

      5. *-rgt-identity 72.7%

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

   5. Simplified 72.7%

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

      1. div-inv 72.7%

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

      2. unpow2 72.7%

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

      3. associate-*l* 82.0%

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

   7. Applied egg-rr 82.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.14 \cdot 10^{+18}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(\left(x - z\right) \cdot \left(x + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 320000000:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 320000000.0) (* (/ x y_m) (/ x 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 <= 320000000.0) {
		tmp = (x / y_m) * (x / 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 <= 320000000.0d0) then
        tmp = (x / y_m) * (x / 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 <= 320000000.0) {
		tmp = (x / y_m) * (x / 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 <= 320000000.0:
		tmp = (x / y_m) * (x / 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 <= 320000000.0)
		tmp = Float64(Float64(x / y_m) * Float64(x / 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 <= 320000000.0)
		tmp = (x / y_m) * (x / 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, 320000000.0], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e8

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. unpow2 35.9%

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

      2. times-frac 37.3%

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

   5. Applied egg-rr 37.3%

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

   if 3.2e8 < y

   1. Initial program 51.9%

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

   3. Taylor expanded in y around inf 55.9%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 320000000:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 240000000:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 240000000.0) (* x (* x (/ 0.5 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 <= 240000000.0) {
		tmp = x * (x * (0.5 / 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 <= 240000000.0d0) then
        tmp = x * (x * (0.5d0 / 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 <= 240000000.0) {
		tmp = x * (x * (0.5 / 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 <= 240000000.0:
		tmp = x * (x * (0.5 / 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 <= 240000000.0)
		tmp = Float64(x * Float64(x * Float64(0.5 / 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 <= 240000000.0)
		tmp = x * (x * (0.5 / 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, 240000000.0], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e8

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. div-inv 35.9%

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

      2. unpow2 35.9%

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

      3. associate-*l* 37.3%

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

      4. *-commutative 37.3%

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

      5. associate-/r* 37.3%

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

      6. metadata-eval 37.3%

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

   5. Applied egg-rr 37.3%

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

   if 2.4e8 < y

   1. Initial program 51.9%

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

   3. Taylor expanded in y around inf 55.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 240000000:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.0% 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 72.5%

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

3. Taylor expanded in y around inf 31.9%

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

4. Final simplification 31.9%

   \[\leadsto y \cdot 0.5 \]
5. 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 2024096 
(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)))