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

Percentage Accurate: 69.5% → 89.6%

Time: 11.9s

Alternatives: 9

Speedup: 1.1×

• 41.4% 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: 69.5% 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: 89.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m - z, y\_m + z, x \cdot x\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.4e+177)
    (/ (fma (- y_m z) (+ y_m z) (* x x)) (* y_m 2.0))
    (* 0.5 (- y_m (* z (/ 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 (y_m <= 1.4e+177) {
		tmp = fma((y_m - z), (y_m + z), (x * x)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (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 (y_m <= 1.4e+177)
		tmp = Float64(fma(Float64(y_m - z), Float64(y_m + z), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.4e+177], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.40000000000000001e177

   1. Initial program 76.3%

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

      1. associate--l+ 76.3%

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

      2. +-commutative 76.3%

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

      3. sqr-neg 76.3%

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

      4. difference-of-squares 77.8%

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

      5. fma-def 81.2%

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

      6. sub-neg 81.2%

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

      7. sub-neg 81.2%

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

      8. remove-double-neg 81.2%

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

   3. Simplified 81.2%

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

   if 1.40000000000000001e177 < y

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 6.4%

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

      1. div-sub 6.4%

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

      2. unpow2 6.4%

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

      3. associate-/l* 89.2%

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

      4. *-inverses 89.2%

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

      5. /-rgt-identity 89.2%

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

   5. Simplified 89.2%

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

      1. unpow2 89.2%

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

      2. add-sqr-sqrt 89.2%

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

      3. times-frac 93.4%

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

   7. Applied egg-rr 93.4%

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

      1. unpow2 93.4%

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

   9. Simplified 93.4%

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

       1. unpow2 93.4%

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

       2. div-inv 93.4%

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

       3. associate-*l* 93.4%

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

       4. times-frac 93.4%

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

       5. *-un-lft-identity 93.4%

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

       6. add-sqr-sqrt 93.4%

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

   11. Applied egg-rr 93.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4e+120)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* 0.5 (- y_m (* z (/ 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 (y_m <= 4e+120) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (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 (y_m <= 4d+120) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z * (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 (y_m <= 4e+120) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (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 y_m <= 4e+120:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z * (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 (y_m <= 4e+120)
		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 / 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 <= 4e+120)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z * (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[y$95$m, 4e+120], 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 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999999e120

   1. Initial program 76.9%

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

   if 3.9999999999999999e120 < y

   1. Initial program 24.7%

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

   3. Taylor expanded in x around 0 27.6%

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

      1. div-sub 27.6%

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

      2. unpow2 27.6%

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

      3. associate-/l* 84.5%

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

      4. *-inverses 84.5%

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

      5. /-rgt-identity 84.5%

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

   5. Simplified 84.5%

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

      1. unpow2 84.5%

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

      2. add-sqr-sqrt 84.5%

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

      3. times-frac 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. unpow2 87.4%

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

   9. Simplified 87.4%

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

       1. unpow2 87.4%

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

       2. div-inv 87.4%

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

       3. associate-*l* 87.4%

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

       4. times-frac 87.4%

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

       5. *-un-lft-identity 87.4%

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

       6. add-sqr-sqrt 87.4%

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

   11. Applied egg-rr 87.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y\_m + z}{\frac{y\_m \cdot 2}{y\_m - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7.5e+75)
    (/ (+ y_m z) (/ (* y_m 2.0) (- y_m z)))
    (/ (* x 0.5) (/ 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 <= 7.5e+75) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else {
		tmp = (x * 0.5) / (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 <= 7.5d+75) then
        tmp = (y_m + z) / ((y_m * 2.0d0) / (y_m - z))
    else
        tmp = (x * 0.5d0) / (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 <= 7.5e+75) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else {
		tmp = (x * 0.5) / (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 <= 7.5e+75:
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z))
	else:
		tmp = (x * 0.5) / (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 <= 7.5e+75)
		tmp = Float64(Float64(y_m + z) / Float64(Float64(y_m * 2.0) / Float64(y_m - z)));
	else
		tmp = Float64(Float64(x * 0.5) / 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 <= 7.5e+75)
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	else
		tmp = (x * 0.5) / (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, 7.5e+75], N[(N[(y$95$m + z), $MachinePrecision] / N[(N[(y$95$m * 2.0), $MachinePrecision] / N[(y$95$m - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999995e75

   1. Initial program 72.7%

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

      1. associate--l+ 72.7%

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

      2. +-commutative 72.7%

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

      3. sqr-neg 72.7%

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

      4. difference-of-squares 75.0%

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

      5. fma-def 75.5%

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

      6. sub-neg 75.5%

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

      7. sub-neg 75.5%

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

      8. remove-double-neg 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate-/l* 78.9%

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

      2. div-inv 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. associate-*r/ 78.9%

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

      2. *-rgt-identity 78.9%

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

      3. +-commutative 78.9%

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

      4. *-commutative 78.9%

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

   9. Simplified 78.9%

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

   if 7.4999999999999995e75 < x

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf 66.8%

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

      1. unpow2 66.8%

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

      2. times-frac 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. div-inv 71.1%

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

      2. metadata-eval 71.1%

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

      3. *-commutative 71.1%

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

      4. clear-num 71.1%

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

      5. un-div-inv 71.2%

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

   7. Applied egg-rr 71.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y + z}{\frac{y \cdot 2}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 3.55e+74)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (/ (* x 0.5) (/ 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 <= 3.55e+74) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * 0.5) / (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 <= 3.55d+74) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * 0.5d0) / (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 <= 3.55e+74) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * 0.5) / (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 <= 3.55e+74:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * 0.5) / (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 <= 3.55e+74)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * 0.5) / 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 <= 3.55e+74)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * 0.5) / (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, 3.55e+74], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.55000000000000001e74

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0 55.6%

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

      1. div-sub 55.6%

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

      2. unpow2 55.6%

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

      3. associate-/l* 75.6%

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

      4. *-inverses 75.6%

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

      5. /-rgt-identity 75.6%

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

   5. Simplified 75.6%

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

      1. unpow2 75.6%

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

      2. add-sqr-sqrt 39.1%

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

      3. times-frac 40.1%

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

   7. Applied egg-rr 40.1%

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

      1. unpow2 40.1%

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

   9. Simplified 40.1%

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

       1. unpow2 40.1%

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

       2. div-inv 40.1%

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

       3. associate-*l* 40.1%

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

       4. times-frac 40.1%

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

       5. *-un-lft-identity 40.1%

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

       6. add-sqr-sqrt 78.8%

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

   11. Applied egg-rr 78.8%

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

   if 3.55000000000000001e74 < x

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf 66.8%

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

      1. unpow2 66.8%

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

      2. times-frac 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. div-inv 71.1%

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

      2. metadata-eval 71.1%

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

      3. *-commutative 71.1%

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

      4. clear-num 71.1%

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

      5. un-div-inv 71.2%

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

   7. Applied egg-rr 71.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+53}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 4.4e+53) (* y_m 0.5) (* 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 <= 4.4e+53) {
		tmp = y_m * 0.5;
	} 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 <= 4.4d+53) then
        tmp = y_m * 0.5d0
    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 <= 4.4e+53) {
		tmp = y_m * 0.5;
	} 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 <= 4.4e+53:
		tmp = y_m * 0.5
	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 <= 4.4e+53)
		tmp = Float64(y_m * 0.5);
	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 <= 4.4e+53)
		tmp = y_m * 0.5;
	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, 4.4e+53], N[(y$95$m * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.39999999999999997e53

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf 42.7%

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

   if 4.39999999999999997e53 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf 66.7%

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

      1. div-inv 66.7%

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

      2. unpow2 66.7%

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

      3. associate-*l* 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-/r* 70.9%

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

      6. metadata-eval 70.9%

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

   5. Applied egg-rr 70.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+53}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+48}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1.32e+48) (* y_m 0.5) (* (/ x y_m) (/ x 2.0)))))

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 <= 1.32e+48) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	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 <= 1.32d+48) then
        tmp = y_m * 0.5d0
    else
        tmp = (x / y_m) * (x / 2.0d0)
    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 <= 1.32e+48) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	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 <= 1.32e+48:
		tmp = y_m * 0.5
	else:
		tmp = (x / y_m) * (x / 2.0)
	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 <= 1.32e+48)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	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 <= 1.32e+48)
		tmp = y_m * 0.5;
	else
		tmp = (x / y_m) * (x / 2.0);
	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, 1.32e+48], N[(y$95$m * 0.5), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e48

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf 42.7%

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

   if 1.32e48 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf 66.7%

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

      1. unpow2 66.7%

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

      2. times-frac 70.9%

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

   5. Applied egg-rr 70.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+42}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 6.4e+42) (* y_m 0.5) (/ x (* y_m (/ 2.0 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 <= 6.4e+42) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (y_m * (2.0 / 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 <= 6.4d+42) then
        tmp = y_m * 0.5d0
    else
        tmp = x / (y_m * (2.0d0 / 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 <= 6.4e+42) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (y_m * (2.0 / 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 <= 6.4e+42:
		tmp = y_m * 0.5
	else:
		tmp = x / (y_m * (2.0 / 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 <= 6.4e+42)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(x / Float64(y_m * Float64(2.0 / 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 <= 6.4e+42)
		tmp = y_m * 0.5;
	else
		tmp = x / (y_m * (2.0 / 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, 6.4e+42], N[(y$95$m * 0.5), $MachinePrecision], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.40000000000000004e42

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf 42.7%

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

   if 6.40000000000000004e42 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf 66.7%

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

      1. unpow2 66.7%

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

      2. times-frac 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. frac-times 66.7%

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

      2. *-commutative 66.7%

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

      3. frac-times 70.9%

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

      4. clear-num 70.9%

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

      5. frac-times 71.0%

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

      6. *-un-lft-identity 71.0%

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

   7. Applied egg-rr 71.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+48}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 8.4e+48) (* y_m 0.5) (/ (* x 0.5) (/ 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 <= 8.4e+48) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x * 0.5) / (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 <= 8.4d+48) then
        tmp = y_m * 0.5d0
    else
        tmp = (x * 0.5d0) / (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 <= 8.4e+48) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x * 0.5) / (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 <= 8.4e+48:
		tmp = y_m * 0.5
	else:
		tmp = (x * 0.5) / (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 <= 8.4e+48)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(Float64(x * 0.5) / 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 <= 8.4e+48)
		tmp = y_m * 0.5;
	else
		tmp = (x * 0.5) / (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, 8.4e+48], N[(y$95$m * 0.5), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.3999999999999994e48

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf 42.7%

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

   if 8.3999999999999994e48 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf 66.7%

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

      1. unpow2 66.7%

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

      2. times-frac 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. div-inv 70.9%

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

      2. metadata-eval 70.9%

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

      3. *-commutative 70.9%

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

      4. clear-num 70.9%

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

      5. un-div-inv 71.0%

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

   7. Applied egg-rr 71.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% 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 1 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 69.7%

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

3. Taylor expanded in y around inf 35.0%

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

4. Final simplification 35.0%

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

Developer target: 99.8% 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 2024026 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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