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

Percentage Accurate: 69.0% → 90.9%

Time: 8.5s

Alternatives: 10

Speedup: 1.1×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y_m - z_m, y_m + z_m, x \cdot x\right)}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m - z_m}{y_m} \cdot \frac{y_m + z_m}{2}\\ \end{array} \end{array} \]

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 (<= y_m 2.9e+143)
    (/ (fma (- y_m z_m) (+ y_m z_m) (* x x)) (* y_m 2.0))
    (* (/ (- y_m z_m) y_m) (/ (+ y_m 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 (y_m <= 2.9e+143) {
		tmp = fma((y_m - z_m), (y_m + z_m), (x * x)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z_m) / y_m) * ((y_m + 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 (y_m <= 2.9e+143)
		tmp = Float64(fma(Float64(y_m - z_m), Float64(y_m + z_m), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z_m) / y_m) * Float64(Float64(y_m + 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[y$95$m, 2.9e+143], N[(N[(N[(y$95$m - z$95$m), $MachinePrecision] * N[(y$95$m + z$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e143

   1. Initial program 80.4%

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

      1. associate--l+ 80.4%

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

      2. +-commutative 80.4%

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

      3. sqr-neg 80.4%

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

      4. difference-of-squares 81.5%

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

      5. fma-def 82.9%

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

      6. sub-neg 82.9%

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

      7. sub-neg 82.9%

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

      8. remove-double-neg 82.9%

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

   3. Simplified 82.9%

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

   if 2.8999999999999998e143 < y

   1. Initial program 13.0%

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

      1. associate--l+ 13.0%

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

      2. +-commutative 13.0%

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

      3. sqr-neg 13.0%

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

      4. difference-of-squares 19.4%

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

      5. fma-def 19.4%

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

      6. sub-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. remove-double-neg 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

      2. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]

Alternative 2: 42.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{y_m + z_m}{-2 \cdot \frac{y_m}{z_m}}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y_m}}}\\ \mathbf{elif}\;x \leq 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y_m}}}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (/ (+ y_m z_m) (* -2.0 (/ y_m z_m)))))
   (*
    y_s
    (if (<= x 2.3e-121)
      (* y_m 0.5)
      (if (<= x 1.25e-27)
        t_0
        (if (<= x 2.9e+25)
          (/ x (/ 2.0 (/ x y_m)))
          (if (<= x 1e+108) t_0 (/ 1.0 (/ 2.0 (* x (/ x y_m)))))))))))

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 t_0 = (y_m + z_m) / (-2.0 * (y_m / z_m));
	double tmp;
	if (x <= 2.3e-121) {
		tmp = y_m * 0.5;
	} else if (x <= 1.25e-27) {
		tmp = t_0;
	} else if (x <= 2.9e+25) {
		tmp = x / (2.0 / (x / y_m));
	} else if (x <= 1e+108) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y_m + z_m) / ((-2.0d0) * (y_m / z_m))
    if (x <= 2.3d-121) then
        tmp = y_m * 0.5d0
    else if (x <= 1.25d-27) then
        tmp = t_0
    else if (x <= 2.9d+25) then
        tmp = x / (2.0d0 / (x / y_m))
    else if (x <= 1d+108) then
        tmp = t_0
    else
        tmp = 1.0d0 / (2.0d0 / (x * (x / y_m)))
    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 t_0 = (y_m + z_m) / (-2.0 * (y_m / z_m));
	double tmp;
	if (x <= 2.3e-121) {
		tmp = y_m * 0.5;
	} else if (x <= 1.25e-27) {
		tmp = t_0;
	} else if (x <= 2.9e+25) {
		tmp = x / (2.0 / (x / y_m));
	} else if (x <= 1e+108) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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):
	t_0 = (y_m + z_m) / (-2.0 * (y_m / z_m))
	tmp = 0
	if x <= 2.3e-121:
		tmp = y_m * 0.5
	elif x <= 1.25e-27:
		tmp = t_0
	elif x <= 2.9e+25:
		tmp = x / (2.0 / (x / y_m))
	elif x <= 1e+108:
		tmp = t_0
	else:
		tmp = 1.0 / (2.0 / (x * (x / y_m)))
	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)
	t_0 = Float64(Float64(y_m + z_m) / Float64(-2.0 * Float64(y_m / z_m)))
	tmp = 0.0
	if (x <= 2.3e-121)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.25e-27)
		tmp = t_0;
	elseif (x <= 2.9e+25)
		tmp = Float64(x / Float64(2.0 / Float64(x / y_m)));
	elseif (x <= 1e+108)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(x * Float64(x / y_m))));
	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)
	t_0 = (y_m + z_m) / (-2.0 * (y_m / z_m));
	tmp = 0.0;
	if (x <= 2.3e-121)
		tmp = y_m * 0.5;
	elseif (x <= 1.25e-27)
		tmp = t_0;
	elseif (x <= 2.9e+25)
		tmp = x / (2.0 / (x / y_m));
	elseif (x <= 1e+108)
		tmp = t_0;
	else
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	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_] := Block[{t$95$0 = N[(N[(y$95$m + z$95$m), $MachinePrecision] / N[(-2.0 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 2.3e-121], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.25e-27], t$95$0, If[LessEqual[x, 2.9e+25], N[(x / N[(2.0 / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+108], t$95$0, N[(1.0 / N[(2.0 / N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 2.30000000000000012e-121

   1. Initial program 71.4%

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

   2. Taylor expanded in y around inf 38.3%

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

   if 2.30000000000000012e-121 < x < 1.25e-27 or 2.8999999999999999e25 < x < 1e108

   1. Initial program 74.4%

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

      1. associate--l+ 74.4%

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

      2. +-commutative 74.4%

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

      3. sqr-neg 74.4%

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

      4. difference-of-squares 77.2%

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

      5. fma-def 77.2%

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

      6. sub-neg 77.2%

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

      7. sub-neg 77.2%

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

      8. remove-double-neg 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in x around 0 59.6%

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

      1. associate-/l* 82.2%

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

      2. div-inv 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. *-commutative 82.2%

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

   8. Simplified 82.2%

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

   9. Taylor expanded in y around 0 54.2%

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

   if 1.25e-27 < x < 2.8999999999999999e25

   1. Initial program 71.7%

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

   2. Taylor expanded in x around inf 41.4%

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

      1. div-inv 41.5%

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

      2. metadata-eval 41.5%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 41.5%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 41.5%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 41.5%

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

      6. associate-*l* 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. associate-*r/ 41.2%

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

      2. metadata-eval 41.2%

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

      3. div-inv 41.2%

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

      4. associate-*r/ 41.4%

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

      5. associate-*l/ 41.2%

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

      6. *-commutative 41.2%

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

      7. clear-num 41.2%

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

      8. frac-times 41.2%

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

      9. *-un-lft-identity 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. associate-*l/ 41.1%

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

      2. associate-/l* 41.4%

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

   8. Applied egg-rr 41.4%

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

   if 1e108 < x

   1. Initial program 60.1%

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

   2. Taylor expanded in x around inf 66.6%

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

      1. div-inv 66.6%

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

      2. metadata-eval 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 66.6%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 66.6%

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

      6. associate-*l* 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. associate-*r/ 85.6%

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

      2. metadata-eval 85.6%

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

      3. div-inv 85.6%

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

      4. associate-*r/ 66.6%

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

      5. associate-*l/ 85.6%

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

      6. clear-num 85.6%

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

      7. clear-num 85.5%

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

      8. frac-times 85.5%

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

      9. metadata-eval 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. clear-num 85.5%

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

      2. frac-times 85.6%

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

      3. metadata-eval 85.6%

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

   8. Applied egg-rr 85.6%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{y + z}{-2 \cdot \frac{y}{z}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq 10^{+108}:\\ \;\;\;\;\frac{y + z}{-2 \cdot \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y}}}\\ \end{array} \]

Alternative 3: 42.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{-126}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{z_m}{y_m} \cdot \frac{\left(-z_m\right) - y_m}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y_m}}}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+110}:\\ \;\;\;\;\frac{y_m + z_m}{-2 \cdot \frac{y_m}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y_m}}}\\ \end{array} \end{array} \]

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 1e-126)
    (* y_m 0.5)
    (if (<= x 1.3e-27)
      (* (/ z_m y_m) (/ (- (- z_m) y_m) 2.0))
      (if (<= x 3.8e+24)
        (/ x (/ 2.0 (/ x y_m)))
        (if (<= x 3.1e+110)
          (/ (+ y_m z_m) (* -2.0 (/ y_m z_m)))
          (/ 1.0 (/ 2.0 (* x (/ x y_m))))))))))

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 <= 1e-126) {
		tmp = y_m * 0.5;
	} else if (x <= 1.3e-27) {
		tmp = (z_m / y_m) * ((-z_m - y_m) / 2.0);
	} else if (x <= 3.8e+24) {
		tmp = x / (2.0 / (x / y_m));
	} else if (x <= 3.1e+110) {
		tmp = (y_m + z_m) / (-2.0 * (y_m / z_m));
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 1d-126) then
        tmp = y_m * 0.5d0
    else if (x <= 1.3d-27) then
        tmp = (z_m / y_m) * ((-z_m - y_m) / 2.0d0)
    else if (x <= 3.8d+24) then
        tmp = x / (2.0d0 / (x / y_m))
    else if (x <= 3.1d+110) then
        tmp = (y_m + z_m) / ((-2.0d0) * (y_m / z_m))
    else
        tmp = 1.0d0 / (2.0d0 / (x * (x / y_m)))
    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 <= 1e-126) {
		tmp = y_m * 0.5;
	} else if (x <= 1.3e-27) {
		tmp = (z_m / y_m) * ((-z_m - y_m) / 2.0);
	} else if (x <= 3.8e+24) {
		tmp = x / (2.0 / (x / y_m));
	} else if (x <= 3.1e+110) {
		tmp = (y_m + z_m) / (-2.0 * (y_m / z_m));
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 1e-126:
		tmp = y_m * 0.5
	elif x <= 1.3e-27:
		tmp = (z_m / y_m) * ((-z_m - y_m) / 2.0)
	elif x <= 3.8e+24:
		tmp = x / (2.0 / (x / y_m))
	elif x <= 3.1e+110:
		tmp = (y_m + z_m) / (-2.0 * (y_m / z_m))
	else:
		tmp = 1.0 / (2.0 / (x * (x / y_m)))
	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 <= 1e-126)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.3e-27)
		tmp = Float64(Float64(z_m / y_m) * Float64(Float64(Float64(-z_m) - y_m) / 2.0));
	elseif (x <= 3.8e+24)
		tmp = Float64(x / Float64(2.0 / Float64(x / y_m)));
	elseif (x <= 3.1e+110)
		tmp = Float64(Float64(y_m + z_m) / Float64(-2.0 * Float64(y_m / z_m)));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(x * Float64(x / y_m))));
	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 <= 1e-126)
		tmp = y_m * 0.5;
	elseif (x <= 1.3e-27)
		tmp = (z_m / y_m) * ((-z_m - y_m) / 2.0);
	elseif (x <= 3.8e+24)
		tmp = x / (2.0 / (x / y_m));
	elseif (x <= 3.1e+110)
		tmp = (y_m + z_m) / (-2.0 * (y_m / z_m));
	else
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	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, 1e-126], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.3e-27], N[(N[(z$95$m / y$95$m), $MachinePrecision] * N[(N[((-z$95$m) - y$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+24], N[(x / N[(2.0 / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+110], N[(N[(y$95$m + z$95$m), $MachinePrecision] / N[(-2.0 * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if x < 9.9999999999999995e-127

   1. Initial program 71.4%

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

   2. Taylor expanded in y around inf 38.3%

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

   if 9.9999999999999995e-127 < x < 1.30000000000000009e-27

   1. Initial program 78.8%

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

      1. associate--l+ 78.8%

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

      2. +-commutative 78.8%

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

      3. sqr-neg 78.8%

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

      4. difference-of-squares 79.7%

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

      5. fma-def 79.7%

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

      6. sub-neg 79.7%

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

      7. sub-neg 79.7%

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

      8. remove-double-neg 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in x around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. times-frac 79.2%

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

   6. Applied egg-rr 79.2%

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

   7. Taylor expanded in y around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. distribute-neg-frac 49.2%

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

   9. Simplified 49.2%

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

   if 1.30000000000000009e-27 < x < 3.80000000000000015e24

   1. Initial program 71.7%

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

   2. Taylor expanded in x around inf 41.4%

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

      1. div-inv 41.5%

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

      2. metadata-eval 41.5%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 41.5%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 41.5%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 41.5%

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

      6. associate-*l* 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. associate-*r/ 41.2%

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

      2. metadata-eval 41.2%

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

      3. div-inv 41.2%

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

      4. associate-*r/ 41.4%

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

      5. associate-*l/ 41.2%

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

      6. *-commutative 41.2%

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

      7. clear-num 41.2%

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

      8. frac-times 41.2%

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

      9. *-un-lft-identity 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. associate-*l/ 41.1%

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

      2. associate-/l* 41.4%

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

   8. Applied egg-rr 41.4%

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

   if 3.80000000000000015e24 < x < 3.10000000000000017e110

   1. Initial program 70.6%

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

      1. associate--l+ 70.6%

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

      2. +-commutative 70.6%

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

      3. sqr-neg 70.6%

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

      4. difference-of-squares 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 75.0%

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

      6. sub-neg 75.0%

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

      7. sub-neg 75.0%

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

      8. remove-double-neg 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 60.1%

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

      1. associate-/l* 85.0%

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

      2. div-inv 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. associate-*r/ 85.0%

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

      2. *-rgt-identity 85.0%

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

      3. *-commutative 85.0%

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

   8. Simplified 85.0%

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

   9. Taylor expanded in y around 0 58.7%

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

   if 3.10000000000000017e110 < x

   1. Initial program 60.1%

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

   2. Taylor expanded in x around inf 66.6%

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

      1. div-inv 66.6%

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

      2. metadata-eval 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 66.6%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 66.6%

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

      6. associate-*l* 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. associate-*r/ 85.6%

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

      2. metadata-eval 85.6%

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

      3. div-inv 85.6%

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

      4. associate-*r/ 66.6%

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

      5. associate-*l/ 85.6%

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

      6. clear-num 85.6%

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

      7. clear-num 85.5%

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

      8. frac-times 85.5%

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

      9. metadata-eval 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. clear-num 85.5%

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

      2. frac-times 85.6%

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

      3. metadata-eval 85.6%

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

   8. Applied egg-rr 85.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{\frac{x}{y} \cdot x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-126}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{\left(-z\right) - y}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+110}:\\ \;\;\;\;\frac{y + z}{-2 \cdot \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y}}}\\ \end{array} \]

Alternative 4: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2.25 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(x \cdot x + y_m \cdot y_m\right) - z_m \cdot z_m}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m - z_m}{y_m} \cdot \frac{y_m + z_m}{2}\\ \end{array} \end{array} \]

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 (<= y_m 2.25e+144)
    (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))
    (* (/ (- y_m z_m) y_m) (/ (+ y_m 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 (y_m <= 2.25e+144) {
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z_m) / y_m) * ((y_m + 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 (y_m <= 2.25d+144) then
        tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)
    else
        tmp = ((y_m - z_m) / y_m) * ((y_m + 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 (y_m <= 2.25e+144) {
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z_m) / y_m) * ((y_m + 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 y_m <= 2.25e+144:
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	else:
		tmp = ((y_m - z_m) / y_m) * ((y_m + 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 (y_m <= 2.25e+144)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z_m) / y_m) * Float64(Float64(y_m + 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 (y_m <= 2.25e+144)
		tmp = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	else
		tmp = ((y_m - z_m) / y_m) * ((y_m + 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[y$95$m, 2.25e+144], 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[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.24999999999999984e144

   1. Initial program 80.4%

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

   if 2.24999999999999984e144 < y

   1. Initial program 13.0%

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

      1. associate--l+ 13.0%

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

      2. +-commutative 13.0%

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

      3. sqr-neg 13.0%

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

      4. difference-of-squares 19.4%

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

      5. fma-def 19.4%

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

      6. sub-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. remove-double-neg 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

      2. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]

Alternative 5: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{y_m - z_m}{y_m} \cdot \frac{y_m + z_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y_m}}}\\ \end{array} \end{array} \]

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 2.15e+111)
    (* (/ (- y_m z_m) y_m) (/ (+ y_m z_m) 2.0))
    (/ 1.0 (/ 2.0 (* x (/ x y_m)))))))

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 <= 2.15e+111) {
		tmp = ((y_m - z_m) / y_m) * ((y_m + z_m) / 2.0);
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.15d+111) then
        tmp = ((y_m - z_m) / y_m) * ((y_m + z_m) / 2.0d0)
    else
        tmp = 1.0d0 / (2.0d0 / (x * (x / y_m)))
    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 <= 2.15e+111) {
		tmp = ((y_m - z_m) / y_m) * ((y_m + z_m) / 2.0);
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.15e+111:
		tmp = ((y_m - z_m) / y_m) * ((y_m + z_m) / 2.0)
	else:
		tmp = 1.0 / (2.0 / (x * (x / y_m)))
	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 <= 2.15e+111)
		tmp = Float64(Float64(Float64(y_m - z_m) / y_m) * Float64(Float64(y_m + z_m) / 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(x * Float64(x / y_m))));
	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 <= 2.15e+111)
		tmp = ((y_m - z_m) / y_m) * ((y_m + z_m) / 2.0);
	else
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	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, 2.15e+111], N[(N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999997e111

   1. Initial program 72.1%

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

      1. associate--l+ 72.1%

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

      2. +-commutative 72.1%

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

      3. sqr-neg 72.1%

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

      4. difference-of-squares 73.8%

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

      5. fma-def 74.7%

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

      6. sub-neg 74.7%

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

      7. sub-neg 74.7%

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

      8. remove-double-neg 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. times-frac 71.5%

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

   6. Applied egg-rr 71.5%

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

   if 2.14999999999999997e111 < x

   1. Initial program 60.1%

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

   2. Taylor expanded in x around inf 66.6%

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

      1. div-inv 66.6%

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

      2. metadata-eval 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 66.6%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 66.6%

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

      6. associate-*l* 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. associate-*r/ 85.6%

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

      2. metadata-eval 85.6%

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

      3. div-inv 85.6%

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

      4. associate-*r/ 66.6%

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

      5. associate-*l/ 85.6%

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

      6. clear-num 85.6%

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

      7. clear-num 85.5%

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

      8. frac-times 85.5%

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

      9. metadata-eval 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. clear-num 85.5%

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

      2. frac-times 85.6%

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

      3. metadata-eval 85.6%

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

   8. Applied egg-rr 85.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y}}}\\ \end{array} \]

Alternative 6: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{y_m + z_m}{\frac{y_m \cdot 2}{y_m - z_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y_m}}}\\ \end{array} \end{array} \]

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 2.8e+110)
    (/ (+ y_m z_m) (/ (* y_m 2.0) (- y_m z_m)))
    (/ 1.0 (/ 2.0 (* x (/ x y_m)))))))

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 <= 2.8e+110) {
		tmp = (y_m + z_m) / ((y_m * 2.0) / (y_m - z_m));
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.8d+110) then
        tmp = (y_m + z_m) / ((y_m * 2.0d0) / (y_m - z_m))
    else
        tmp = 1.0d0 / (2.0d0 / (x * (x / y_m)))
    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 <= 2.8e+110) {
		tmp = (y_m + z_m) / ((y_m * 2.0) / (y_m - z_m));
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.8e+110:
		tmp = (y_m + z_m) / ((y_m * 2.0) / (y_m - z_m))
	else:
		tmp = 1.0 / (2.0 / (x * (x / y_m)))
	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 <= 2.8e+110)
		tmp = Float64(Float64(y_m + z_m) / Float64(Float64(y_m * 2.0) / Float64(y_m - z_m)));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(x * Float64(x / y_m))));
	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 <= 2.8e+110)
		tmp = (y_m + z_m) / ((y_m * 2.0) / (y_m - z_m));
	else
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	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, 2.8e+110], N[(N[(y$95$m + z$95$m), $MachinePrecision] / N[(N[(y$95$m * 2.0), $MachinePrecision] / N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999987e110

   1. Initial program 72.1%

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

      1. associate--l+ 72.1%

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

      2. +-commutative 72.1%

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

      3. sqr-neg 72.1%

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

      4. difference-of-squares 73.8%

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

      5. fma-def 74.7%

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

      6. sub-neg 74.7%

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

      7. sub-neg 74.7%

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

      8. remove-double-neg 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around 0 50.7%

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

      1. associate-/l* 71.5%

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

      2. div-inv 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*r/ 71.5%

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

      2. *-rgt-identity 71.5%

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

      3. *-commutative 71.5%

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

   8. Simplified 71.5%

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

   if 2.79999999999999987e110 < x

   1. Initial program 60.1%

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

   2. Taylor expanded in x around inf 66.6%

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

      1. div-inv 66.6%

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

      2. metadata-eval 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 66.6%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 66.6%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 66.6%

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

      6. associate-*l* 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. associate-*r/ 85.6%

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

      2. metadata-eval 85.6%

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

      3. div-inv 85.6%

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

      4. associate-*r/ 66.6%

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

      5. associate-*l/ 85.6%

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

      6. clear-num 85.6%

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

      7. clear-num 85.5%

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

      8. frac-times 85.5%

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

      9. metadata-eval 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. clear-num 85.5%

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

      2. frac-times 85.6%

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

      3. metadata-eval 85.6%

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

   8. Applied egg-rr 85.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{y + z}{\frac{y \cdot 2}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y}}}\\ \end{array} \]

Alternative 7: 43.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+83}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y_m}}}\\ \end{array} \end{array} \]

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 2.15e+83) (* y_m 0.5) (/ 1.0 (/ 2.0 (* x (/ x y_m)))))))

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 <= 2.15e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.15d+83) then
        tmp = y_m * 0.5d0
    else
        tmp = 1.0d0 / (2.0d0 / (x * (x / y_m)))
    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 <= 2.15e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	}
	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 <= 2.15e+83:
		tmp = y_m * 0.5
	else:
		tmp = 1.0 / (2.0 / (x * (x / y_m)))
	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 <= 2.15e+83)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(x * Float64(x / y_m))));
	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 <= 2.15e+83)
		tmp = y_m * 0.5;
	else
		tmp = 1.0 / (2.0 / (x * (x / y_m)));
	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, 2.15e+83], N[(y$95$m * 0.5), $MachinePrecision], N[(1.0 / N[(2.0 / N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.15e83

   1. Initial program 71.6%

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

   2. Taylor expanded in y around inf 37.8%

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

   if 2.15e83 < x

   1. Initial program 65.0%

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

   2. Taylor expanded in x around inf 59.7%

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

      1. div-inv 59.7%

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

      2. metadata-eval 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 59.7%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 59.7%

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

      6. associate-*l* 75.2%

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

   4. Applied egg-rr 75.2%

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

      1. associate-*r/ 75.2%

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

      2. metadata-eval 75.2%

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

      3. div-inv 75.2%

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

      4. associate-*r/ 59.7%

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

      5. associate-*l/ 75.2%

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

      6. clear-num 75.2%

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

      7. clear-num 75.1%

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

      8. frac-times 75.1%

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

      9. metadata-eval 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. clear-num 75.1%

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

      2. frac-times 75.2%

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

      3. metadata-eval 75.2%

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

   8. Applied egg-rr 75.2%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+83}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{x \cdot \frac{x}{y}}}\\ \end{array} \]

Alternative 8: 43.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y_m}\right)\\ \end{array} \end{array} \]

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 1.5e+83) (* y_m 0.5) (* x (* x (/ 0.5 y_m))))))

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 <= 1.5e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (0.5 / y_m));
	}
	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 <= 1.5d+83) then
        tmp = y_m * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y_m))
    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 <= 1.5e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (0.5 / y_m));
	}
	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 <= 1.5e+83:
		tmp = y_m * 0.5
	else:
		tmp = x * (x * (0.5 / y_m))
	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 <= 1.5e+83)
		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

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 <= 1.5e+83)
		tmp = y_m * 0.5;
	else
		tmp = x * (x * (0.5 / y_m));
	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, 1.5e+83], 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 < 1.5e83

   1. Initial program 71.6%

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

   2. Taylor expanded in y around inf 37.8%

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

   if 1.5e83 < x

   1. Initial program 65.0%

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

   2. Taylor expanded in x around inf 59.7%

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

      1. div-inv 59.7%

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

      2. metadata-eval 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 59.7%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 59.7%

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

      6. associate-*l* 75.2%

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

   4. Applied egg-rr 75.2%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 9: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+83}:\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y_m}}}\\ \end{array} \end{array} \]

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 7e+83) (* y_m 0.5) (/ x (/ 2.0 (/ x y_m))))))

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 <= 7e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (2.0 / (x / y_m));
	}
	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 <= 7d+83) then
        tmp = y_m * 0.5d0
    else
        tmp = x / (2.0d0 / (x / y_m))
    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 <= 7e+83) {
		tmp = y_m * 0.5;
	} else {
		tmp = x / (2.0 / (x / y_m));
	}
	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 <= 7e+83:
		tmp = y_m * 0.5
	else:
		tmp = x / (2.0 / (x / y_m))
	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 <= 7e+83)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(x / Float64(2.0 / Float64(x / y_m)));
	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 <= 7e+83)
		tmp = y_m * 0.5;
	else
		tmp = x / (2.0 / (x / y_m));
	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, 7e+83], N[(y$95$m * 0.5), $MachinePrecision], N[(x / N[(2.0 / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999954e83

   1. Initial program 71.6%

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

   2. Taylor expanded in y around inf 37.8%

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

   if 6.99999999999999954e83 < x

   1. Initial program 65.0%

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

   2. Taylor expanded in x around inf 59.7%

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

      1. div-inv 59.7%

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

      2. metadata-eval 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 59.7%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 59.7%

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{0.5}{y}} \]

      5. unpow2 59.7%

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

      6. associate-*l* 75.2%

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

   4. Applied egg-rr 75.2%

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

      1. associate-*r/ 75.2%

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

      2. metadata-eval 75.2%

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

      3. div-inv 75.2%

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

      4. associate-*r/ 59.7%

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

      5. associate-*l/ 75.2%

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

      6. *-commutative 75.2%

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

      7. clear-num 75.1%

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

      8. frac-times 75.1%

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

      9. *-un-lft-identity 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-*l/ 75.1%

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

      2. associate-/l* 75.2%

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

   8. Applied egg-rr 75.2%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+83}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y}}}\\ \end{array} \]

Alternative 10: 34.4% 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 70.6%

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

2. Taylor expanded in y around inf 33.0%

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

3. Final simplification 33.0%

   \[\leadsto y \cdot 0.5 \]

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 2023318 
(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)))