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

Percentage Accurate: 69.2% → 93.5%

Time: 11.3s

Alternatives: 11

Speedup: 0.8×

• 42.1% 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: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 5 \cdot 10^{+169}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{2 \cdot y_m}}\right)}^{2} + \left(0.5 \cdot \left(z_m + y_m\right)\right) \cdot \left(1 - \frac{z_m}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 5e+169)
    (+
     (pow (/ x (sqrt (* 2.0 y_m))) 2.0)
     (* (* 0.5 (+ z_m y_m)) (- 1.0 (/ z_m y_m))))
    (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+169) {
		tmp = pow((x / sqrt((2.0 * y_m))), 2.0) + ((0.5 * (z_m + y_m)) * (1.0 - (z_m / y_m)));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 5d+169) then
        tmp = ((x / sqrt((2.0d0 * y_m))) ** 2.0d0) + ((0.5d0 * (z_m + y_m)) * (1.0d0 - (z_m / y_m)))
    else
        tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+169) {
		tmp = Math.pow((x / Math.sqrt((2.0 * y_m))), 2.0) + ((0.5 * (z_m + y_m)) * (1.0 - (z_m / y_m)));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 5e+169:
		tmp = math.pow((x / math.sqrt((2.0 * y_m))), 2.0) + ((0.5 * (z_m + y_m)) * (1.0 - (z_m / y_m)))
	else:
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+169)
		tmp = Float64((Float64(x / sqrt(Float64(2.0 * y_m))) ^ 2.0) + Float64(Float64(0.5 * Float64(z_m + y_m)) * Float64(1.0 - Float64(z_m / y_m))));
	else
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 5e+169)
		tmp = ((x / sqrt((2.0 * y_m))) ^ 2.0) + ((0.5 * (z_m + y_m)) * (1.0 - (z_m / y_m)));
	else
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+169], N[(N[Power[N[(x / N[Sqrt[N[(2.0 * y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 * N[(z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000017e169

   1. Initial program 76.5%

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

      1. associate--l+ 76.5%

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

      2. +-commutative 76.5%

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

      3. sqr-neg 76.5%

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

      4. difference-of-squares 76.5%

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

      5. fma-def 76.5%

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

      6. sub-neg 76.5%

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

      7. sub-neg 76.5%

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

      8. remove-double-neg 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around 0 75.2%

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

   7. Applied egg-rr 50.5%

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

      1. fma-udef 50.5%

         \[\leadsto \color{blue}{\frac{x}{\sqrt{y \cdot 2}} \cdot \frac{x}{\sqrt{y \cdot 2}} + \frac{y + z}{\frac{y}{y - z}} \cdot 0.5} \]

      2. unpow2 50.5%

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

      3. *-commutative 50.5%

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

      4. *-commutative 50.5%

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

      5. associate-/l* 40.6%

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

      6. associate-*r/ 50.5%

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

      7. associate-*r* 50.5%

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

      8. +-commutative 50.5%

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

      9. div-sub 50.5%

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

      10. *-inverses 50.5%

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

   9. Simplified 50.5%

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

   if 5.00000000000000017e169 < (*.f64 z z)

   1. Initial program 68.4%

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

      1. flip-+ 43.6%

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

      2. clear-num 43.6%

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

      3. pow2 43.6%

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

      4. pow2 43.6%

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

      5. pow2 43.6%

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

      6. pow2 43.6%

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

      7. pow-prod-up 43.6%

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

      8. metadata-eval 43.6%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 43.6%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 43.6%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 43.6%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 43.6%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 43.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 71.8%

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

      1. remove-double-div 71.8%

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

      2. unpow2 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. difference-of-squares 85.0%

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

      2. times-frac 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+169}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{2 \cdot y}}\right)}^{2} + \left(0.5 \cdot \left(z + y\right)\right) \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-102}:\\ \;\;\;\;\frac{z_m + y_m}{y_m} \cdot \left(0.5 \cdot \left(y_m - z_m\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y_m - z_m, z_m + y_m, x \cdot x\right)}{2 \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* x x) 1e-102)
    (* (/ (+ z_m y_m) y_m) (* 0.5 (- y_m z_m)))
    (if (<= (* x x) 4e+279)
      (/ (fma (- y_m z_m) (+ z_m y_m) (* x x)) (* 2.0 y_m))
      (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((x * x) <= 1e-102) {
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	} else if ((x * x) <= 4e+279) {
		tmp = fma((y_m - z_m), (z_m + y_m), (x * x)) / (2.0 * y_m);
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-102)
		tmp = Float64(Float64(Float64(z_m + y_m) / y_m) * Float64(0.5 * Float64(y_m - z_m)));
	elseif (Float64(x * x) <= 4e+279)
		tmp = Float64(fma(Float64(y_m - z_m), Float64(z_m + y_m), Float64(x * x)) / Float64(2.0 * y_m));
	else
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[N[(x * x), $MachinePrecision], 1e-102], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+279], N[(N[(N[(y$95$m - z$95$m), $MachinePrecision] * N[(z$95$m + y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999933e-103

   1. Initial program 73.5%

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

      1. associate--l+ 73.5%

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

      2. +-commutative 73.5%

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

      3. sqr-neg 73.5%

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

      4. difference-of-squares 74.6%

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

      5. fma-def 74.6%

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

      6. sub-neg 74.6%

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

      7. sub-neg 74.6%

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

      8. remove-double-neg 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 72.1%

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

      1. times-frac 95.5%

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

      2. div-inv 95.5%

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

      3. metadata-eval 95.5%

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

   7. Applied egg-rr 95.5%

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

   if 9.99999999999999933e-103 < (*.f64 x x) < 4.00000000000000023e279

   1. Initial program 86.1%

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

      1. associate--l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. sqr-neg 86.1%

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

      4. difference-of-squares 87.6%

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

      5. fma-def 87.7%

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

      6. sub-neg 87.7%

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

      7. sub-neg 87.7%

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

      8. remove-double-neg 87.7%

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

   3. Simplified 87.7%

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

   if 4.00000000000000023e279 < (*.f64 x x)

   1. Initial program 59.7%

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

      1. flip-+ 4.3%

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

      2. clear-num 4.3%

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

      3. pow2 4.3%

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

      4. pow2 4.3%

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

      5. pow2 4.3%

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

      6. pow2 4.3%

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

      7. pow-prod-up 4.3%

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

      8. metadata-eval 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 4.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 61.1%

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

      1. remove-double-div 61.1%

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

      2. unpow2 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. difference-of-squares 78.8%

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

      2. times-frac 92.1%

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-102}:\\ \;\;\;\;\frac{z + y}{y} \cdot \left(0.5 \cdot \left(y - z\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, z + y, x \cdot x\right)}{2 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;0.5 \cdot \left(\frac{z_m + y_m}{\frac{y_m}{y_m - z_m}} + \frac{{x}^{2}}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* x x) 4e+279)
    (* 0.5 (+ (/ (+ z_m y_m) (/ y_m (- y_m z_m))) (/ (pow x 2.0) y_m)))
    (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((x * x) <= 4e+279) {
		tmp = 0.5 * (((z_m + y_m) / (y_m / (y_m - z_m))) + (pow(x, 2.0) / y_m));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * x) <= 4d+279) then
        tmp = 0.5d0 * (((z_m + y_m) / (y_m / (y_m - z_m))) + ((x ** 2.0d0) / y_m))
    else
        tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if ((x * x) <= 4e+279) {
		tmp = 0.5 * (((z_m + y_m) / (y_m / (y_m - z_m))) + (Math.pow(x, 2.0) / y_m));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if (x * x) <= 4e+279:
		tmp = 0.5 * (((z_m + y_m) / (y_m / (y_m - z_m))) + (math.pow(x, 2.0) / y_m))
	else:
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 4e+279)
		tmp = Float64(0.5 * Float64(Float64(Float64(z_m + y_m) / Float64(y_m / Float64(y_m - z_m))) + Float64((x ^ 2.0) / y_m)));
	else
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if ((x * x) <= 4e+279)
		tmp = 0.5 * (((z_m + y_m) / (y_m / (y_m - z_m))) + ((x ^ 2.0) / y_m));
	else
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[N[(x * x), $MachinePrecision], 4e+279], N[(0.5 * N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] / N[(y$95$m / N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000023e279

   1. Initial program 78.7%

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

      1. associate--l+ 78.7%

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

      2. +-commutative 78.7%

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

      3. sqr-neg 78.7%

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

      4. difference-of-squares 80.0%

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

      5. fma-def 80.0%

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

      6. sub-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. remove-double-neg 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. distribute-lft-out 76.8%

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

      2. associate-/l* 95.5%

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

   7. Applied egg-rr 95.5%

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

   if 4.00000000000000023e279 < (*.f64 x x)

   1. Initial program 59.7%

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

      1. flip-+ 4.3%

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

      2. clear-num 4.3%

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

      3. pow2 4.3%

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

      4. pow2 4.3%

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

      5. pow2 4.3%

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

      6. pow2 4.3%

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

      7. pow-prod-up 4.3%

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

      8. metadata-eval 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 4.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 61.1%

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

      1. remove-double-div 61.1%

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

      2. unpow2 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. difference-of-squares 78.8%

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

      2. times-frac 92.1%

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;0.5 \cdot \left(\frac{z + y}{\frac{y}{y - z}} + \frac{{x}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-102}:\\ \;\;\;\;\frac{z_m + y_m}{y_m} \cdot \left(0.5 \cdot \left(y_m - z_m\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{\left(x \cdot x + y_m \cdot y_m\right) - z_m \cdot z_m}{2 \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* x x) 1e-102)
    (* (/ (+ z_m y_m) y_m) (* 0.5 (- y_m z_m)))
    (if (<= (* x x) 4e+279)
      (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* 2.0 y_m))
      (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((x * x) <= 1e-102) {
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	} else if ((x * x) <= 4e+279) {
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (2.0 * y_m);
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * x) <= 1d-102) then
        tmp = ((z_m + y_m) / y_m) * (0.5d0 * (y_m - z_m))
    else if ((x * x) <= 4d+279) then
        tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (2.0d0 * y_m)
    else
        tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if ((x * x) <= 1e-102) {
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	} else if ((x * x) <= 4e+279) {
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (2.0 * y_m);
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if (x * x) <= 1e-102:
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m))
	elif (x * x) <= 4e+279:
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (2.0 * y_m)
	else:
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-102)
		tmp = Float64(Float64(Float64(z_m + y_m) / y_m) * Float64(0.5 * Float64(y_m - z_m)));
	elseif (Float64(x * x) <= 4e+279)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(2.0 * y_m));
	else
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if ((x * x) <= 1e-102)
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	elseif ((x * x) <= 4e+279)
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (2.0 * y_m);
	else
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[N[(x * x), $MachinePrecision], 1e-102], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+279], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999933e-103

   1. Initial program 73.5%

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

      1. associate--l+ 73.5%

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

      2. +-commutative 73.5%

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

      3. sqr-neg 73.5%

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

      4. difference-of-squares 74.6%

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

      5. fma-def 74.6%

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

      6. sub-neg 74.6%

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

      7. sub-neg 74.6%

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

      8. remove-double-neg 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 72.1%

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

      1. times-frac 95.5%

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

      2. div-inv 95.5%

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

      3. metadata-eval 95.5%

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

   7. Applied egg-rr 95.5%

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

   if 9.99999999999999933e-103 < (*.f64 x x) < 4.00000000000000023e279

   1. Initial program 86.1%

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

   if 4.00000000000000023e279 < (*.f64 x x)

   1. Initial program 59.7%

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

      1. flip-+ 4.3%

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

      2. clear-num 4.3%

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

      3. pow2 4.3%

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

      4. pow2 4.3%

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

      5. pow2 4.3%

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

      6. pow2 4.3%

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

      7. pow-prod-up 4.3%

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

      8. metadata-eval 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 4.3%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 4.3%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 4.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 61.1%

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

      1. remove-double-div 61.1%

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

      2. unpow2 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. difference-of-squares 78.8%

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

      2. times-frac 92.1%

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-102}:\\ \;\;\;\;\frac{z + y}{y} \cdot \left(0.5 \cdot \left(y - z\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{2 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \mathbf{elif}\;y_m \leq 1.52 \cdot 10^{+172}:\\ \;\;\;\;\left(\left(z_m + y_m\right) \cdot \left(y_m - z_m\right)\right) \cdot \frac{0.5}{y_m}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.6e-22)
    (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0))
    (if (<= y_m 1.52e+172)
      (* (* (+ z_m y_m) (- y_m z_m)) (/ 0.5 y_m))
      (* y_m 0.5)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3.6e-22) {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	} else if (y_m <= 1.52e+172) {
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 3.6d-22) then
        tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0d0)
    else if (y_m <= 1.52d+172) then
        tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5d0 / y_m)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (y_m <= 3.6e-22) {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	} else if (y_m <= 1.52e+172) {
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if y_m <= 3.6e-22:
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0)
	elif y_m <= 1.52e+172:
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (y_m <= 3.6e-22)
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	elseif (y_m <= 1.52e+172)
		tmp = Float64(Float64(Float64(z_m + y_m) * Float64(y_m - z_m)) * Float64(0.5 / y_m));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 3.6e-22)
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	elseif (y_m <= 1.52e+172)
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 3.6e-22], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.52e+172], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 3.5999999999999998e-22

   1. Initial program 77.9%

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

      1. flip-+ 46.1%

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

      2. clear-num 46.1%

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

      3. pow2 46.1%

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

      4. pow2 46.1%

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

      5. pow2 46.1%

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

      6. pow2 46.1%

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

      7. pow-prod-up 46.1%

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

      8. metadata-eval 46.1%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 46.1%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 46.1%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 46.0%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 46.0%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 46.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 67.8%

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

      1. remove-double-div 68.2%

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

      2. unpow2 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. difference-of-squares 73.2%

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

      2. times-frac 79.7%

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

   9. Applied egg-rr 79.7%

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

   if 3.5999999999999998e-22 < y < 1.5200000000000001e172

   1. Initial program 85.6%

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

      1. associate--l+ 85.6%

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

      2. +-commutative 85.6%

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

      3. sqr-neg 85.6%

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

      4. difference-of-squares 85.6%

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

      5. fma-def 85.8%

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

      6. sub-neg 85.8%

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

      7. sub-neg 85.8%

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

      8. remove-double-neg 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. div-inv 53.8%

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

      2. metadata-eval 53.8%

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

      3. div-inv 53.8%

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

      4. clear-num 53.8%

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

   7. Applied egg-rr 53.8%

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

   if 1.5200000000000001e172 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around inf 69.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+172}:\\ \;\;\;\;\left(\left(z + y\right) \cdot \left(y - z\right)\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-90}:\\ \;\;\;\;\frac{z_m + y_m}{y_m} \cdot \left(0.5 \cdot \left(y_m - z_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z_m + x}{y_m} \cdot \frac{x - z_m}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* x x) 1e-90)
    (* (/ (+ z_m y_m) y_m) (* 0.5 (- y_m z_m)))
    (* (/ (+ z_m x) y_m) (/ (- x z_m) 2.0)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((x * x) <= 1e-90) {
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * x) <= 1d-90) then
        tmp = ((z_m + y_m) / y_m) * (0.5d0 * (y_m - z_m))
    else
        tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if ((x * x) <= 1e-90) {
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	} else {
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if (x * x) <= 1e-90:
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m))
	else:
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-90)
		tmp = Float64(Float64(Float64(z_m + y_m) / y_m) * Float64(0.5 * Float64(y_m - z_m)));
	else
		tmp = Float64(Float64(Float64(z_m + x) / y_m) * Float64(Float64(x - z_m) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if ((x * x) <= 1e-90)
		tmp = ((z_m + y_m) / y_m) * (0.5 * (y_m - z_m));
	else
		tmp = ((z_m + x) / y_m) * ((x - z_m) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[N[(x * x), $MachinePrecision], 1e-90], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999995e-91

   1. Initial program 73.4%

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

      1. associate--l+ 73.4%

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

      2. +-commutative 73.4%

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

      3. sqr-neg 73.4%

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

      4. difference-of-squares 74.4%

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

      5. fma-def 74.4%

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

      6. sub-neg 74.4%

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

      7. sub-neg 74.4%

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

      8. remove-double-neg 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in x around 0 71.2%

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

      1. times-frac 94.8%

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

      2. div-inv 94.8%

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

      3. metadata-eval 94.8%

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

   7. Applied egg-rr 94.8%

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

   if 9.99999999999999995e-91 < (*.f64 x x)

   1. Initial program 73.6%

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

      1. flip-+ 32.2%

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

      2. clear-num 32.2%

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

      3. pow2 32.2%

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

      4. pow2 32.2%

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

      5. pow2 32.2%

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

      6. pow2 32.2%

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

      7. pow-prod-up 32.2%

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

      8. metadata-eval 32.2%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{\color{blue}{4}} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      9. pow2 32.2%

         \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)}} - z \cdot z}{y \cdot 2} \]

      10. pow2 32.2%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{2} \cdot \color{blue}{{y}^{2}}}} - z \cdot z}{y \cdot 2} \]

      11. pow-prod-up 32.2%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}}}} - z \cdot z}{y \cdot 2} \]

      12. metadata-eval 32.2%

          \[\leadsto \frac{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{\color{blue}{4}}}} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 32.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{x}^{2} - {y}^{2}}{{x}^{4} - {y}^{4}}}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in x around inf 68.8%

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

      1. remove-double-div 68.8%

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

      2. unpow2 68.8%

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

   7. Applied egg-rr 68.8%

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

      1. difference-of-squares 77.4%

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

      2. times-frac 85.7%

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

   9. Applied egg-rr 85.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-90}:\\ \;\;\;\;\frac{z + y}{y} \cdot \left(0.5 \cdot \left(y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x}{y} \cdot \frac{x - z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(z_m + y_m\right) \cdot \left(y_m - z_m\right)\right) \cdot \frac{0.5}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y_m}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= x 1.95e+89)
    (* (* (+ z_m y_m) (- y_m z_m)) (/ 0.5 y_m))
    (/ (* x 0.5) (/ y_m x)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (x <= 1.95e+89) {
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 1.95d+89) then
        tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5d0 / y_m)
    else
        tmp = (x * 0.5d0) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (x <= 1.95e+89) {
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if x <= 1.95e+89:
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m)
	else:
		tmp = (x * 0.5) / (y_m / x)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (x <= 1.95e+89)
		tmp = Float64(Float64(Float64(z_m + y_m) * Float64(y_m - z_m)) * Float64(0.5 / y_m));
	else
		tmp = Float64(Float64(x * 0.5) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (x <= 1.95e+89)
		tmp = ((z_m + y_m) * (y_m - z_m)) * (0.5 / y_m);
	else
		tmp = (x * 0.5) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[x, 1.95e+89], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y$95$m), $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 < 1.95000000000000005e89

   1. Initial program 75.0%

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

      1. associate--l+ 75.0%

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

      2. +-commutative 75.0%

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

      3. sqr-neg 75.0%

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

      4. difference-of-squares 76.7%

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

      5. fma-def 77.6%

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

      6. sub-neg 77.6%

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

      7. sub-neg 77.6%

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

      8. remove-double-neg 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around 0 56.0%

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

      1. div-inv 56.0%

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

      2. metadata-eval 56.0%

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

      3. div-inv 56.0%

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

      4. clear-num 56.0%

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

   7. Applied egg-rr 56.0%

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

   if 1.95000000000000005e89 < x

   1. Initial program 65.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. unpow2 73.5%

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

      2. times-frac 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. *-commutative 85.9%

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

      2. clear-num 85.8%

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

      3. un-div-inv 85.9%

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

      4. div-inv 85.9%

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

      5. metadata-eval 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(z + y\right) \cdot \left(y - z\right)\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= x 7.6e-44) (* y_m 0.5) (* (/ x y_m) (/ x 2.0)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (x <= 7.6e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 7.6d-44) then
        tmp = y_m * 0.5d0
    else
        tmp = (x / y_m) * (x / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (x <= 7.6e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if x <= 7.6e-44:
		tmp = y_m * 0.5
	else:
		tmp = (x / y_m) * (x / 2.0)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (x <= 7.6e-44)
		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

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (x <= 7.6e-44)
		tmp = y_m * 0.5;
	else
		tmp = (x / y_m) * (x / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[x, 7.6e-44], 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 < 7.6000000000000002e-44

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 32.7%

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

   if 7.6000000000000002e-44 < x

   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 inf 54.0%

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

      1. unpow2 54.0%

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

      2. times-frac 61.0%

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

   5. Applied egg-rr 61.0%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-44}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y_m \cdot \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= x 5.5e-44) (* y_m 0.5) (/ x (* y_m (/ 2.0 x))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (x <= 5.5e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 5.5d-44) then
        tmp = y_m * 0.5d0
    else
        tmp = x / (y_m * (2.0d0 / x))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (x <= 5.5e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if x <= 5.5e-44:
		tmp = y_m * 0.5
	else:
		tmp = x / (y_m * (2.0 / x))
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (x <= 5.5e-44)
		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

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (x <= 5.5e-44)
		tmp = y_m * 0.5;
	else
		tmp = x / (y_m * (2.0 / x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[x, 5.5e-44], 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 < 5.49999999999999993e-44

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 32.7%

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

   if 5.49999999999999993e-44 < x

   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 inf 54.0%

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

      1. unpow2 54.0%

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

      2. times-frac 61.0%

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

   5. Applied egg-rr 61.0%

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

      1. *-commutative 61.0%

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

      2. clear-num 61.0%

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

      3. frac-times 61.0%

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

      4. *-un-lft-identity 61.0%

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

   7. Applied egg-rr 61.0%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-44}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y_m}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= x 5.2e-44) (* y_m 0.5) (/ (* x 0.5) (/ y_m x)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (x <= 5.2e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 5.2d-44) then
        tmp = y_m * 0.5d0
    else
        tmp = (x * 0.5d0) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (x <= 5.2e-44) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if x <= 5.2e-44:
		tmp = y_m * 0.5
	else:
		tmp = (x * 0.5) / (y_m / x)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (x <= 5.2e-44)
		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

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (x <= 5.2e-44)
		tmp = y_m * 0.5;
	else
		tmp = (x * 0.5) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[x, 5.2e-44], 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 < 5.1999999999999996e-44

   1. Initial program 73.8%

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

   3. Taylor expanded in y around inf 32.7%

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

   if 5.1999999999999996e-44 < x

   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 inf 54.0%

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

      1. unpow2 54.0%

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

      2. times-frac 61.0%

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

   5. Applied egg-rr 61.0%

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

      1. *-commutative 61.0%

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

      2. clear-num 60.9%

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

      3. un-div-inv 61.1%

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

      4. div-inv 61.1%

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

      5. metadata-eval 61.1%

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

   7. Applied egg-rr 61.1%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(y_m \cdot 0.5\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m) :precision binary64 (* y_s (* y_m 0.5)))

C

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

Fortran

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

Java

z_m = Math.abs(z);
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_m) {
	return y_s * (y_m * 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in y around inf 28.0%

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

4. Final simplification 28.0%

   \[\leadsto y \cdot 0.5 \]
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 2024019 
(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)))