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

Percentage Accurate: 68.9% → 96.0%

Time: 13.2s

Alternatives: 11

Speedup: 1.1×

• 41.1% 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: 68.9% 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: 96.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ t_1 := \mathsf{hypot}\left(x, y\_m + z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-132} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t\_1}{\frac{y\_m}{t\_1}}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)))
        (t_1 (hypot x (+ y_m z))))
   (*
    y_s
    (if (or (<= t_0 -1e-132) (not (<= t_0 INFINITY)))
      (* 0.5 (- y_m (* z (/ z y_m))))
      (* 0.5 (/ t_1 (/ y_m t_1)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double t_1 = hypot(x, (y_m + z));
	double tmp;
	if ((t_0 <= -1e-132) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * (t_1 / (y_m / t_1));
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double t_1 = Math.hypot(x, (y_m + z));
	double tmp;
	if ((t_0 <= -1e-132) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * (t_1 / (y_m / t_1));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	t_1 = math.hypot(x, (y_m + z))
	tmp = 0
	if (t_0 <= -1e-132) or not (t_0 <= math.inf):
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = 0.5 * (t_1 / (y_m / t_1))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	t_1 = hypot(x, Float64(y_m + z))
	tmp = 0.0
	if ((t_0 <= -1e-132) || !(t_0 <= Inf))
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(0.5 * Float64(t_1 / Float64(y_m / t_1)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	t_1 = hypot(x, (y_m + z));
	tmp = 0.0;
	if ((t_0 <= -1e-132) || ~((t_0 <= Inf)))
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = 0.5 * (t_1 / (y_m / t_1));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[x ^ 2 + N[(y$95$m + z), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -1e-132], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$1 / N[(y$95$m / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -9.9999999999999999e-133 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.9%

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

      1. remove-double-neg 61.9%

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

      2. distribute-lft-neg-out 61.9%

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

      3. distribute-frac-neg2 61.9%

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

      4. distribute-frac-neg 61.9%

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

      5. neg-mul-1 61.9%

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

      6. distribute-lft-neg-out 61.9%

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

      7. *-commutative 61.9%

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

      8. distribute-lft-neg-in 61.9%

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

      9. times-frac 61.9%

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

      10. metadata-eval 61.9%

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

      11. metadata-eval 61.9%

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

      12. associate--l+ 61.9%

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

      13. fma-define 67.9%

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

   3. Simplified 67.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 43.1%

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

      1. div-sub 43.1%

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

      2. unpow2 43.1%

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

      3. associate-/l* 60.4%

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

      4. *-inverses 60.4%

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

      5. *-rgt-identity 60.4%

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

   7. Simplified 60.4%

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

      1. unpow2 60.4%

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

      2. associate-/l* 67.7%

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

   9. Applied egg-rr 67.7%

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

   if -9.9999999999999999e-133 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 77.3%

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

      1. remove-double-neg 77.3%

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

      2. distribute-lft-neg-out 77.3%

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

      3. distribute-frac-neg2 77.3%

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

      4. distribute-frac-neg 77.3%

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

      5. neg-mul-1 77.3%

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

      6. distribute-lft-neg-out 77.3%

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

      7. *-commutative 77.3%

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

      8. distribute-lft-neg-in 77.3%

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

      9. times-frac 77.3%

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

      10. metadata-eval 77.3%

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

      11. metadata-eval 77.3%

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

      12. associate--l+ 77.3%

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

      13. fma-define 77.3%

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

   3. Simplified 77.3%

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

      1. prod-diff 59.2%

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

      2. fma-neg 59.2%

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

      3. difference-of-squares 59.2%

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

      4. fma-define 59.3%

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

      5. pow2 59.3%

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

   6. Applied egg-rr 59.3%

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

      1. add-sqr-sqrt 45.5%

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

      2. *-un-lft-identity 45.5%

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

      3. times-frac 45.5%

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

   8. Applied egg-rr 65.6%

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

      1. /-rgt-identity 65.6%

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

      2. clear-num 65.5%

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

      3. un-div-inv 65.6%

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

   10. Applied egg-rr 65.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-132} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{hypot}\left(x, y + z\right)}{\frac{y}{\mathsf{hypot}\left(x, y + z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m - z \cdot z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2e+151)
    (* 0.5 (/ (fma x x (- (* y_m y_m) (* z z))) y_m))
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+151) {
		tmp = 0.5 * (fma(x, x, ((y_m * y_m) - (z * z))) / y_m);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e+151)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y_m * y_m) - Float64(z * z))) / y_m));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2e+151], N[(0.5 * N[(N[(x * x + N[(N[(y$95$m * y$95$m), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000003e151

   1. Initial program 78.9%

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

      1. remove-double-neg 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-frac-neg2 78.9%

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

      4. distribute-frac-neg 78.9%

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

      5. neg-mul-1 78.9%

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

      6. distribute-lft-neg-out 78.9%

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

      7. *-commutative 78.9%

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

      8. distribute-lft-neg-in 78.9%

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

      9. times-frac 78.9%

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

      10. metadata-eval 78.9%

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

      11. metadata-eval 78.9%

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

      12. associate--l+ 78.9%

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

      13. fma-define 82.5%

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

   3. Simplified 82.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   if 2.00000000000000003e151 < y

   1. Initial program 12.9%

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

      1. remove-double-neg 12.9%

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

      2. distribute-lft-neg-out 12.9%

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

      3. distribute-frac-neg2 12.9%

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

      4. distribute-frac-neg 12.9%

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

      5. neg-mul-1 12.9%

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

      6. distribute-lft-neg-out 12.9%

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

      7. *-commutative 12.9%

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

      8. distribute-lft-neg-in 12.9%

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

      9. times-frac 12.9%

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

      10. metadata-eval 12.9%

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

      11. metadata-eval 12.9%

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

      12. associate--l+ 12.9%

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

      13. fma-define 12.9%

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

   3. Simplified 12.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 12.9%

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

      1. div-sub 12.9%

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

      2. unpow2 12.9%

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

      3. associate-/l* 77.0%

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

      4. *-inverses 77.0%

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

      5. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. unpow2 77.0%

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

      2. associate-/l* 92.4%

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

   9. Applied egg-rr 92.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-119}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+92} \lor \neg \left(z \leq 4.2 \cdot 10^{+134}\right) \land z \leq 9 \cdot 10^{+158}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y_m)))))
   (*
    y_s
    (if (<= z 1e-214)
      t_0
      (if (<= z 1.25e-119)
        (* y_m 0.5)
        (if (<= z 1.7e-14)
          t_0
          (if (<= z 1.95e+32)
            (* y_m 0.5)
            (if (<= z 1.35e+41)
              t_0
              (if (<= z 8e+62)
                (* z (* z (/ -0.5 y_m)))
                (if (or (<= z 6.5e+92)
                        (and (not (<= z 4.2e+134)) (<= z 9e+158)))
                  (* y_m 0.5)
                  (* z (/ (/ z -2.0) y_m))))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = x * (x * (0.5 / y_m));
	double tmp;
	if (z <= 1e-214) {
		tmp = t_0;
	} else if (z <= 1.25e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 1.7e-14) {
		tmp = t_0;
	} else if (z <= 1.95e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 1.35e+41) {
		tmp = t_0;
	} else if (z <= 8e+62) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 6.5e+92) || (!(z <= 4.2e+134) && (z <= 9e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y_m))
    if (z <= 1d-214) then
        tmp = t_0
    else if (z <= 1.25d-119) then
        tmp = y_m * 0.5d0
    else if (z <= 1.7d-14) then
        tmp = t_0
    else if (z <= 1.95d+32) then
        tmp = y_m * 0.5d0
    else if (z <= 1.35d+41) then
        tmp = t_0
    else if (z <= 8d+62) then
        tmp = z * (z * ((-0.5d0) / y_m))
    else if ((z <= 6.5d+92) .or. (.not. (z <= 4.2d+134)) .and. (z <= 9d+158)) then
        tmp = y_m * 0.5d0
    else
        tmp = z * ((z / (-2.0d0)) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = x * (x * (0.5 / y_m));
	double tmp;
	if (z <= 1e-214) {
		tmp = t_0;
	} else if (z <= 1.25e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 1.7e-14) {
		tmp = t_0;
	} else if (z <= 1.95e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 1.35e+41) {
		tmp = t_0;
	} else if (z <= 8e+62) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 6.5e+92) || (!(z <= 4.2e+134) && (z <= 9e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = x * (x * (0.5 / y_m))
	tmp = 0
	if z <= 1e-214:
		tmp = t_0
	elif z <= 1.25e-119:
		tmp = y_m * 0.5
	elif z <= 1.7e-14:
		tmp = t_0
	elif z <= 1.95e+32:
		tmp = y_m * 0.5
	elif z <= 1.35e+41:
		tmp = t_0
	elif z <= 8e+62:
		tmp = z * (z * (-0.5 / y_m))
	elif (z <= 6.5e+92) or (not (z <= 4.2e+134) and (z <= 9e+158)):
		tmp = y_m * 0.5
	else:
		tmp = z * ((z / -2.0) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y_m)))
	tmp = 0.0
	if (z <= 1e-214)
		tmp = t_0;
	elseif (z <= 1.25e-119)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 1.7e-14)
		tmp = t_0;
	elseif (z <= 1.95e+32)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 1.35e+41)
		tmp = t_0;
	elseif (z <= 8e+62)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	elseif ((z <= 6.5e+92) || (!(z <= 4.2e+134) && (z <= 9e+158)))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(Float64(z / -2.0) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = x * (x * (0.5 / y_m));
	tmp = 0.0;
	if (z <= 1e-214)
		tmp = t_0;
	elseif (z <= 1.25e-119)
		tmp = y_m * 0.5;
	elseif (z <= 1.7e-14)
		tmp = t_0;
	elseif (z <= 1.95e+32)
		tmp = y_m * 0.5;
	elseif (z <= 1.35e+41)
		tmp = t_0;
	elseif (z <= 8e+62)
		tmp = z * (z * (-0.5 / y_m));
	elseif ((z <= 6.5e+92) || (~((z <= 4.2e+134)) && (z <= 9e+158)))
		tmp = y_m * 0.5;
	else
		tmp = z * ((z / -2.0) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 1e-214], t$95$0, If[LessEqual[z, 1.25e-119], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 1.7e-14], t$95$0, If[LessEqual[z, 1.95e+32], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 1.35e+41], t$95$0, If[LessEqual[z, 8e+62], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 6.5e+92], And[N[Not[LessEqual[z, 4.2e+134]], $MachinePrecision], LessEqual[z, 9e+158]]], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(N[(z / -2.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 9.99999999999999913e-215 or 1.24999999999999998e-119 < z < 1.70000000000000001e-14 or 1.95e32 < z < 1.35e41

   1. Initial program 73.1%

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

      1. clear-num 73.1%

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

      2. inv-pow 73.1%

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

      3. associate-/l* 72.7%

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

      4. add-sqr-sqrt 72.7%

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

      5. pow2 72.7%

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

      6. hypot-define 72.7%

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

      7. pow2 72.7%

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

   4. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 72.7%

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

      2. associate-*r/ 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in x around inf 38.7%

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

      1. associate-*r/ 38.7%

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

   9. Simplified 38.7%

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

       1. associate-/r/ 38.8%

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

       2. *-commutative 38.8%

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

       3. metadata-eval 38.8%

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

       4. div-inv 38.8%

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

       5. clear-num 38.8%

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

       6. unpow2 38.8%

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

       7. associate-*r* 40.5%

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

   11. Applied egg-rr 40.5%

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

   if 9.99999999999999913e-215 < z < 1.24999999999999998e-119 or 1.70000000000000001e-14 < z < 1.95e32 or 8.00000000000000028e62 < z < 6.49999999999999999e92 or 4.2000000000000002e134 < z < 9.00000000000000092e158

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. *-commutative 66.9%

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

   5. Simplified 66.9%

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

   if 1.35e41 < z < 8.00000000000000028e62

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

   5. Simplified 50.6%

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

      1. associate-/l* 50.6%

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

      2. pow2 50.6%

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

      3. associate-*l* 50.2%

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

   7. Applied egg-rr 50.2%

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

   if 6.49999999999999999e92 < z < 4.2000000000000002e134 or 9.00000000000000092e158 < z

   1. Initial program 59.1%

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

   3. Taylor expanded in z around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l/ 73.2%

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

   5. Simplified 73.2%

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

      1. associate-/l* 73.2%

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

      2. pow2 73.2%

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

      3. associate-*l* 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. clear-num 79.8%

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

      2. un-div-inv 79.9%

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

      3. div-inv 79.9%

         \[\leadsto z \cdot \frac{z}{\color{blue}{y \cdot \frac{1}{-0.5}}} \]

      4. metadata-eval 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. *-commutative 79.9%

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

       2. associate-/r* 79.9%

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

   11. Simplified 79.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-214}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-119}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+92} \lor \neg \left(z \leq 4.2 \cdot 10^{+134}\right) \land z \leq 9 \cdot 10^{+158}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x}{y\_m} \cdot \frac{x}{2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+92} \lor \neg \left(z \leq 8 \cdot 10^{+133}\right) \land z \leq 9 \cdot 10^{+158}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= z 6e-215)
      t_0
      (if (<= z 1.02e-119)
        (* y_m 0.5)
        (if (<= z 6.2e-15)
          t_0
          (if (<= z 2e+32)
            (* y_m 0.5)
            (if (<= z 4e+37)
              (* x (* x (/ 0.5 y_m)))
              (if (<= z 1.32e+60)
                (* z (* z (/ -0.5 y_m)))
                (if (or (<= z 1.3e+92) (and (not (<= z 8e+133)) (<= z 9e+158)))
                  (* y_m 0.5)
                  (* z (/ (/ z -2.0) y_m))))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x / y_m) * (x / 2.0);
	double tmp;
	if (z <= 6e-215) {
		tmp = t_0;
	} else if (z <= 1.02e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 6.2e-15) {
		tmp = t_0;
	} else if (z <= 2e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 4e+37) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 1.32e+60) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 1.3e+92) || (!(z <= 8e+133) && (z <= 9e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y_m) * (x / 2.0d0)
    if (z <= 6d-215) then
        tmp = t_0
    else if (z <= 1.02d-119) then
        tmp = y_m * 0.5d0
    else if (z <= 6.2d-15) then
        tmp = t_0
    else if (z <= 2d+32) then
        tmp = y_m * 0.5d0
    else if (z <= 4d+37) then
        tmp = x * (x * (0.5d0 / y_m))
    else if (z <= 1.32d+60) then
        tmp = z * (z * ((-0.5d0) / y_m))
    else if ((z <= 1.3d+92) .or. (.not. (z <= 8d+133)) .and. (z <= 9d+158)) then
        tmp = y_m * 0.5d0
    else
        tmp = z * ((z / (-2.0d0)) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x / y_m) * (x / 2.0);
	double tmp;
	if (z <= 6e-215) {
		tmp = t_0;
	} else if (z <= 1.02e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 6.2e-15) {
		tmp = t_0;
	} else if (z <= 2e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 4e+37) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 1.32e+60) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 1.3e+92) || (!(z <= 8e+133) && (z <= 9e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (x / y_m) * (x / 2.0)
	tmp = 0
	if z <= 6e-215:
		tmp = t_0
	elif z <= 1.02e-119:
		tmp = y_m * 0.5
	elif z <= 6.2e-15:
		tmp = t_0
	elif z <= 2e+32:
		tmp = y_m * 0.5
	elif z <= 4e+37:
		tmp = x * (x * (0.5 / y_m))
	elif z <= 1.32e+60:
		tmp = z * (z * (-0.5 / y_m))
	elif (z <= 1.3e+92) or (not (z <= 8e+133) and (z <= 9e+158)):
		tmp = y_m * 0.5
	else:
		tmp = z * ((z / -2.0) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (z <= 6e-215)
		tmp = t_0;
	elseif (z <= 1.02e-119)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 6.2e-15)
		tmp = t_0;
	elseif (z <= 2e+32)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 4e+37)
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	elseif (z <= 1.32e+60)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	elseif ((z <= 1.3e+92) || (!(z <= 8e+133) && (z <= 9e+158)))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(Float64(z / -2.0) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (z <= 6e-215)
		tmp = t_0;
	elseif (z <= 1.02e-119)
		tmp = y_m * 0.5;
	elseif (z <= 6.2e-15)
		tmp = t_0;
	elseif (z <= 2e+32)
		tmp = y_m * 0.5;
	elseif (z <= 4e+37)
		tmp = x * (x * (0.5 / y_m));
	elseif (z <= 1.32e+60)
		tmp = z * (z * (-0.5 / y_m));
	elseif ((z <= 1.3e+92) || (~((z <= 8e+133)) && (z <= 9e+158)))
		tmp = y_m * 0.5;
	else
		tmp = z * ((z / -2.0) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 6e-215], t$95$0, If[LessEqual[z, 1.02e-119], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 6.2e-15], t$95$0, If[LessEqual[z, 2e+32], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 4e+37], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.32e+60], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.3e+92], And[N[Not[LessEqual[z, 8e+133]], $MachinePrecision], LessEqual[z, 9e+158]]], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(N[(z / -2.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if z < 6.00000000000000051e-215 or 1.02e-119 < z < 6.1999999999999998e-15

   1. Initial program 72.8%

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

      1. clear-num 72.8%

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

      2. inv-pow 72.8%

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

      3. associate-/l* 72.4%

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

      4. add-sqr-sqrt 72.4%

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

      5. pow2 72.4%

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

      6. hypot-define 72.4%

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

      7. pow2 72.4%

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

   4. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 72.4%

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

      2. associate-*r/ 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in x around inf 38.0%

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

      1. associate-*r/ 38.0%

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

   9. Simplified 38.0%

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

       1. clear-num 38.1%

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

       2. unpow2 38.1%

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

       3. *-commutative 38.1%

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

       4. times-frac 39.9%

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

   11. Applied egg-rr 39.9%

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

   if 6.00000000000000051e-215 < z < 1.02e-119 or 6.1999999999999998e-15 < z < 2.00000000000000011e32 or 1.32e60 < z < 1.2999999999999999e92 or 8.0000000000000002e133 < z < 9.00000000000000092e158

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. *-commutative 66.9%

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

   5. Simplified 66.9%

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

   if 2.00000000000000011e32 < z < 3.99999999999999982e37

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l* 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. pow2 100.0%

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

      6. hypot-define 100.0%

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

      7. pow2 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/r/ 100.0%

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

       2. *-commutative 100.0%

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

       3. metadata-eval 100.0%

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

       4. div-inv 100.0%

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

       5. clear-num 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 3.99999999999999982e37 < z < 1.32e60

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

   5. Simplified 50.6%

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

      1. associate-/l* 50.6%

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

      2. pow2 50.6%

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

      3. associate-*l* 50.2%

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

   7. Applied egg-rr 50.2%

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

   if 1.2999999999999999e92 < z < 8.0000000000000002e133 or 9.00000000000000092e158 < z

   1. Initial program 59.1%

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

   3. Taylor expanded in z around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l/ 73.2%

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

   5. Simplified 73.2%

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

      1. associate-/l* 73.2%

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

      2. pow2 73.2%

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

      3. associate-*l* 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. clear-num 79.8%

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

      2. un-div-inv 79.9%

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

      3. div-inv 79.9%

         \[\leadsto z \cdot \frac{z}{\color{blue}{y \cdot \frac{1}{-0.5}}} \]

      4. metadata-eval 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. *-commutative 79.9%

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

       2. associate-/r* 79.9%

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

   11. Simplified 79.9%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+92} \lor \neg \left(z \leq 8 \cdot 10^{+133}\right) \land z \leq 9 \cdot 10^{+158}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-214}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+91} \lor \neg \left(z \leq 3.8 \cdot 10^{+134}\right) \land z \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 2.3e-214)
    (* (/ x y_m) (/ x 2.0))
    (if (<= z 1.02e-119)
      (* y_m 0.5)
      (if (<= z 2.7e-14)
        (/ (* x 0.5) (/ y_m x))
        (if (<= z 1.95e+32)
          (* y_m 0.5)
          (if (<= z 1.45e+40)
            (* x (* x (/ 0.5 y_m)))
            (if (<= z 2.85e+60)
              (* z (* z (/ -0.5 y_m)))
              (if (or (<= z 3.3e+91)
                      (and (not (<= z 3.8e+134)) (<= z 1.1e+158)))
                (* y_m 0.5)
                (* z (/ (/ z -2.0) y_m)))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.3e-214) {
		tmp = (x / y_m) * (x / 2.0);
	} else if (z <= 1.02e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 2.7e-14) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (z <= 1.95e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 1.45e+40) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 2.85e+60) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 3.3e+91) || (!(z <= 3.8e+134) && (z <= 1.1e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.3d-214) then
        tmp = (x / y_m) * (x / 2.0d0)
    else if (z <= 1.02d-119) then
        tmp = y_m * 0.5d0
    else if (z <= 2.7d-14) then
        tmp = (x * 0.5d0) / (y_m / x)
    else if (z <= 1.95d+32) then
        tmp = y_m * 0.5d0
    else if (z <= 1.45d+40) then
        tmp = x * (x * (0.5d0 / y_m))
    else if (z <= 2.85d+60) then
        tmp = z * (z * ((-0.5d0) / y_m))
    else if ((z <= 3.3d+91) .or. (.not. (z <= 3.8d+134)) .and. (z <= 1.1d+158)) then
        tmp = y_m * 0.5d0
    else
        tmp = z * ((z / (-2.0d0)) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.3e-214) {
		tmp = (x / y_m) * (x / 2.0);
	} else if (z <= 1.02e-119) {
		tmp = y_m * 0.5;
	} else if (z <= 2.7e-14) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (z <= 1.95e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 1.45e+40) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 2.85e+60) {
		tmp = z * (z * (-0.5 / y_m));
	} else if ((z <= 3.3e+91) || (!(z <= 3.8e+134) && (z <= 1.1e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 2.3e-214:
		tmp = (x / y_m) * (x / 2.0)
	elif z <= 1.02e-119:
		tmp = y_m * 0.5
	elif z <= 2.7e-14:
		tmp = (x * 0.5) / (y_m / x)
	elif z <= 1.95e+32:
		tmp = y_m * 0.5
	elif z <= 1.45e+40:
		tmp = x * (x * (0.5 / y_m))
	elif z <= 2.85e+60:
		tmp = z * (z * (-0.5 / y_m))
	elif (z <= 3.3e+91) or (not (z <= 3.8e+134) and (z <= 1.1e+158)):
		tmp = y_m * 0.5
	else:
		tmp = z * ((z / -2.0) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 2.3e-214)
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	elseif (z <= 1.02e-119)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 2.7e-14)
		tmp = Float64(Float64(x * 0.5) / Float64(y_m / x));
	elseif (z <= 1.95e+32)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 1.45e+40)
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	elseif (z <= 2.85e+60)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	elseif ((z <= 3.3e+91) || (!(z <= 3.8e+134) && (z <= 1.1e+158)))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(Float64(z / -2.0) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 2.3e-214)
		tmp = (x / y_m) * (x / 2.0);
	elseif (z <= 1.02e-119)
		tmp = y_m * 0.5;
	elseif (z <= 2.7e-14)
		tmp = (x * 0.5) / (y_m / x);
	elseif (z <= 1.95e+32)
		tmp = y_m * 0.5;
	elseif (z <= 1.45e+40)
		tmp = x * (x * (0.5 / y_m));
	elseif (z <= 2.85e+60)
		tmp = z * (z * (-0.5 / y_m));
	elseif ((z <= 3.3e+91) || (~((z <= 3.8e+134)) && (z <= 1.1e+158)))
		tmp = y_m * 0.5;
	else
		tmp = z * ((z / -2.0) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 2.3e-214], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.02e-119], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 2.7e-14], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+32], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 1.45e+40], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.85e+60], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.3e+91], And[N[Not[LessEqual[z, 3.8e+134]], $MachinePrecision], LessEqual[z, 1.1e+158]]], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(N[(z / -2.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]

Derivation

1. Split input into 6 regimes

2. if z < 2.30000000000000011e-214

   1. Initial program 70.5%

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

      1. clear-num 70.4%

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

      2. inv-pow 70.4%

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

      3. associate-/l* 70.0%

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

      4. add-sqr-sqrt 70.0%

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

      5. pow2 70.0%

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

      6. hypot-define 70.0%

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

      7. pow2 70.0%

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

   4. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 70.0%

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

      2. associate-*r/ 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in x around inf 35.7%

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

      1. associate-*r/ 35.7%

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

   9. Simplified 35.7%

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

       1. clear-num 35.7%

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

       2. unpow2 35.7%

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

       3. *-commutative 35.7%

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

       4. times-frac 37.8%

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

   11. Applied egg-rr 37.8%

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

   if 2.30000000000000011e-214 < z < 1.02e-119 or 2.6999999999999999e-14 < z < 1.95e32 or 2.84999999999999989e60 < z < 3.30000000000000017e91 or 3.79999999999999998e134 < z < 1.1000000000000001e158

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. *-commutative 66.9%

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

   5. Simplified 66.9%

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

   if 1.02e-119 < z < 2.6999999999999999e-14

   1. Initial program 90.5%

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

      1. clear-num 90.5%

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

      2. inv-pow 90.5%

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

      3. associate-/l* 90.6%

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

      4. add-sqr-sqrt 90.6%

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

      5. pow2 90.6%

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

      6. hypot-define 90.6%

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

      7. pow2 90.6%

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

   4. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.6%

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

      2. associate-*r/ 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in x around inf 56.3%

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

      1. associate-*r/ 56.3%

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

   9. Simplified 56.3%

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

       1. clear-num 56.4%

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

       2. unpow2 56.4%

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

       3. *-commutative 56.4%

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

       4. times-frac 56.3%

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

   11. Applied egg-rr 56.3%

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

       1. *-commutative 56.3%

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

       2. clear-num 56.3%

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

       3. un-div-inv 56.3%

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

       4. div-inv 56.3%

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

       5. metadata-eval 56.3%

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

   13. Applied egg-rr 56.3%

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

   if 1.95e32 < z < 1.45000000000000009e40

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l* 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. pow2 100.0%

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

      6. hypot-define 100.0%

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

      7. pow2 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/r/ 100.0%

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

       2. *-commutative 100.0%

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

       3. metadata-eval 100.0%

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

       4. div-inv 100.0%

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

       5. clear-num 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 1.45000000000000009e40 < z < 2.84999999999999989e60

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

   5. Simplified 50.6%

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

      1. associate-/l* 50.6%

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

      2. pow2 50.6%

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

      3. associate-*l* 50.2%

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

   7. Applied egg-rr 50.2%

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

   if 3.30000000000000017e91 < z < 3.79999999999999998e134 or 1.1000000000000001e158 < z

   1. Initial program 59.1%

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

   3. Taylor expanded in z around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l/ 73.2%

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

   5. Simplified 73.2%

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

      1. associate-/l* 73.2%

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

      2. pow2 73.2%

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

      3. associate-*l* 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. clear-num 79.8%

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

      2. un-div-inv 79.9%

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

      3. div-inv 79.9%

         \[\leadsto z \cdot \frac{z}{\color{blue}{y \cdot \frac{1}{-0.5}}} \]

      4. metadata-eval 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. *-commutative 79.9%

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

       2. associate-/r* 79.9%

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

   11. Simplified 79.9%

       \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{-2}}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-214}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+91} \lor \neg \left(z \leq 3.8 \cdot 10^{+134}\right) \land z \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-120}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+65}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot -0.5\right)}{y\_m}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+92} \lor \neg \left(z \leq 2.05 \cdot 10^{+133}\right) \land z \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 5e-215)
    (* (/ x y_m) (/ x 2.0))
    (if (<= z 8.6e-120)
      (* y_m 0.5)
      (if (<= z 6e-15)
        (/ (* x 0.5) (/ y_m x))
        (if (<= z 2e+32)
          (* y_m 0.5)
          (if (<= z 2.1e+36)
            (* x (* x (/ 0.5 y_m)))
            (if (<= z 2.65e+65)
              (/ (* z (* z -0.5)) y_m)
              (if (or (<= z 2.6e+92)
                      (and (not (<= z 2.05e+133)) (<= z 1.1e+158)))
                (* y_m 0.5)
                (* z (/ (/ z -2.0) y_m)))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 5e-215) {
		tmp = (x / y_m) * (x / 2.0);
	} else if (z <= 8.6e-120) {
		tmp = y_m * 0.5;
	} else if (z <= 6e-15) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (z <= 2e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 2.1e+36) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 2.65e+65) {
		tmp = (z * (z * -0.5)) / y_m;
	} else if ((z <= 2.6e+92) || (!(z <= 2.05e+133) && (z <= 1.1e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5d-215) then
        tmp = (x / y_m) * (x / 2.0d0)
    else if (z <= 8.6d-120) then
        tmp = y_m * 0.5d0
    else if (z <= 6d-15) then
        tmp = (x * 0.5d0) / (y_m / x)
    else if (z <= 2d+32) then
        tmp = y_m * 0.5d0
    else if (z <= 2.1d+36) then
        tmp = x * (x * (0.5d0 / y_m))
    else if (z <= 2.65d+65) then
        tmp = (z * (z * (-0.5d0))) / y_m
    else if ((z <= 2.6d+92) .or. (.not. (z <= 2.05d+133)) .and. (z <= 1.1d+158)) then
        tmp = y_m * 0.5d0
    else
        tmp = z * ((z / (-2.0d0)) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 5e-215) {
		tmp = (x / y_m) * (x / 2.0);
	} else if (z <= 8.6e-120) {
		tmp = y_m * 0.5;
	} else if (z <= 6e-15) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (z <= 2e+32) {
		tmp = y_m * 0.5;
	} else if (z <= 2.1e+36) {
		tmp = x * (x * (0.5 / y_m));
	} else if (z <= 2.65e+65) {
		tmp = (z * (z * -0.5)) / y_m;
	} else if ((z <= 2.6e+92) || (!(z <= 2.05e+133) && (z <= 1.1e+158))) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 5e-215:
		tmp = (x / y_m) * (x / 2.0)
	elif z <= 8.6e-120:
		tmp = y_m * 0.5
	elif z <= 6e-15:
		tmp = (x * 0.5) / (y_m / x)
	elif z <= 2e+32:
		tmp = y_m * 0.5
	elif z <= 2.1e+36:
		tmp = x * (x * (0.5 / y_m))
	elif z <= 2.65e+65:
		tmp = (z * (z * -0.5)) / y_m
	elif (z <= 2.6e+92) or (not (z <= 2.05e+133) and (z <= 1.1e+158)):
		tmp = y_m * 0.5
	else:
		tmp = z * ((z / -2.0) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 5e-215)
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	elseif (z <= 8.6e-120)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 6e-15)
		tmp = Float64(Float64(x * 0.5) / Float64(y_m / x));
	elseif (z <= 2e+32)
		tmp = Float64(y_m * 0.5);
	elseif (z <= 2.1e+36)
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	elseif (z <= 2.65e+65)
		tmp = Float64(Float64(z * Float64(z * -0.5)) / y_m);
	elseif ((z <= 2.6e+92) || (!(z <= 2.05e+133) && (z <= 1.1e+158)))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(Float64(z / -2.0) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 5e-215)
		tmp = (x / y_m) * (x / 2.0);
	elseif (z <= 8.6e-120)
		tmp = y_m * 0.5;
	elseif (z <= 6e-15)
		tmp = (x * 0.5) / (y_m / x);
	elseif (z <= 2e+32)
		tmp = y_m * 0.5;
	elseif (z <= 2.1e+36)
		tmp = x * (x * (0.5 / y_m));
	elseif (z <= 2.65e+65)
		tmp = (z * (z * -0.5)) / y_m;
	elseif ((z <= 2.6e+92) || (~((z <= 2.05e+133)) && (z <= 1.1e+158)))
		tmp = y_m * 0.5;
	else
		tmp = z * ((z / -2.0) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 5e-215], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.6e-120], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 6e-15], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+32], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z, 2.1e+36], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.65e+65], N[(N[(z * N[(z * -0.5), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[Or[LessEqual[z, 2.6e+92], And[N[Not[LessEqual[z, 2.05e+133]], $MachinePrecision], LessEqual[z, 1.1e+158]]], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(N[(z / -2.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]

Derivation

1. Split input into 6 regimes

2. if z < 4.99999999999999956e-215

   1. Initial program 70.5%

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

      1. clear-num 70.4%

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

      2. inv-pow 70.4%

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

      3. associate-/l* 70.0%

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

      4. add-sqr-sqrt 70.0%

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

      5. pow2 70.0%

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

      6. hypot-define 70.0%

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

      7. pow2 70.0%

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

   4. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 70.0%

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

      2. associate-*r/ 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in x around inf 35.7%

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

      1. associate-*r/ 35.7%

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

   9. Simplified 35.7%

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

       1. clear-num 35.7%

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

       2. unpow2 35.7%

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

       3. *-commutative 35.7%

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

       4. times-frac 37.8%

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

   11. Applied egg-rr 37.8%

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

   if 4.99999999999999956e-215 < z < 8.59999999999999964e-120 or 6e-15 < z < 2.00000000000000011e32 or 2.65000000000000011e65 < z < 2.5999999999999999e92 or 2.05000000000000002e133 < z < 1.1000000000000001e158

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf 66.9%

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

      1. *-commutative 66.9%

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

   5. Simplified 66.9%

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

   if 8.59999999999999964e-120 < z < 6e-15

   1. Initial program 90.5%

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

      1. clear-num 90.5%

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

      2. inv-pow 90.5%

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

      3. associate-/l* 90.6%

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

      4. add-sqr-sqrt 90.6%

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

      5. pow2 90.6%

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

      6. hypot-define 90.6%

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

      7. pow2 90.6%

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

   4. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 90.6%

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

      2. associate-*r/ 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in x around inf 56.3%

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

      1. associate-*r/ 56.3%

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

   9. Simplified 56.3%

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

       1. clear-num 56.4%

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

       2. unpow2 56.4%

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

       3. *-commutative 56.4%

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

       4. times-frac 56.3%

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

   11. Applied egg-rr 56.3%

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

       1. *-commutative 56.3%

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

       2. clear-num 56.3%

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

       3. un-div-inv 56.3%

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

       4. div-inv 56.3%

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

       5. metadata-eval 56.3%

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

   13. Applied egg-rr 56.3%

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

   if 2.00000000000000011e32 < z < 2.10000000000000004e36

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l* 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. pow2 100.0%

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

      6. hypot-define 100.0%

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

      7. pow2 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/r/ 100.0%

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

       2. *-commutative 100.0%

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

       3. metadata-eval 100.0%

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

       4. div-inv 100.0%

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

       5. clear-num 100.0%

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

       6. unpow2 100.0%

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

       7. associate-*r* 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 2.10000000000000004e36 < z < 2.65000000000000011e65

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

   5. Simplified 50.6%

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

      1. associate-/l* 50.6%

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

      2. pow2 50.6%

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

      3. associate-*l* 50.2%

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

   7. Applied egg-rr 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-*r/ 50.2%

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

      3. associate-*l/ 50.6%

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

   9. Applied egg-rr 50.6%

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

   if 2.5999999999999999e92 < z < 2.05000000000000002e133 or 1.1000000000000001e158 < z

   1. Initial program 59.1%

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

   3. Taylor expanded in z around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l/ 73.2%

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

   5. Simplified 73.2%

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

      1. associate-/l* 73.2%

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

      2. pow2 73.2%

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

      3. associate-*l* 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. clear-num 79.8%

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

      2. un-div-inv 79.9%

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

      3. div-inv 79.9%

         \[\leadsto z \cdot \frac{z}{\color{blue}{y \cdot \frac{1}{-0.5}}} \]

      4. metadata-eval 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. *-commutative 79.9%

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

       2. associate-/r* 79.9%

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

   11. Simplified 79.9%

       \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{-2}}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-120}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+65}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot -0.5\right)}{y}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+92} \lor \neg \left(z \leq 2.05 \cdot 10^{+133}\right) \land z \leq 1.1 \cdot 10^{+158}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.65e+121)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.65e+121) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.65d+121) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.65e+121) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.65e+121:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.65e+121)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.65e+121)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.65e+121], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.65e121

   1. Initial program 78.9%

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

   if 3.65e121 < y

   1. Initial program 18.9%

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

      1. remove-double-neg 18.9%

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

      2. distribute-lft-neg-out 18.9%

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

      3. distribute-frac-neg2 18.9%

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

      4. distribute-frac-neg 18.9%

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

      5. neg-mul-1 18.9%

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

      6. distribute-lft-neg-out 18.9%

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

      7. *-commutative 18.9%

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

      8. distribute-lft-neg-in 18.9%

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

      9. times-frac 18.9%

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

      10. metadata-eval 18.9%

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

      11. metadata-eval 18.9%

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

      12. associate--l+ 18.9%

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

      13. fma-define 21.4%

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

   3. Simplified 21.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 21.4%

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

      1. div-sub 21.4%

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

      2. unpow2 21.4%

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

      3. associate-/l* 79.3%

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

      4. *-inverses 79.3%

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

      5. *-rgt-identity 79.3%

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

   7. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. associate-/l* 93.1%

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

   9. Applied egg-rr 93.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.25e+172)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (/ (* x 0.5) (/ y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.25e+172) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.25d+172) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * 0.5d0) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.25e+172) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * 0.5) / (y_m / x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.25e+172:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * 0.5) / (y_m / x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.25e+172)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * 0.5) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.25e+172)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * 0.5) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.25e+172], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e172

   1. Initial program 71.1%

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

      1. remove-double-neg 71.1%

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

      2. distribute-lft-neg-out 71.1%

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

      3. distribute-frac-neg2 71.1%

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

      4. distribute-frac-neg 71.1%

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

      5. neg-mul-1 71.1%

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

      6. distribute-lft-neg-out 71.1%

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

      7. *-commutative 71.1%

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

      8. distribute-lft-neg-in 71.1%

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

      9. times-frac 71.1%

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

      10. metadata-eval 71.1%

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

      11. metadata-eval 71.1%

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

      12. associate--l+ 71.1%

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

      13. fma-define 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 49.1%

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

      1. div-sub 49.2%

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

      2. unpow2 49.2%

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

      3. associate-/l* 67.5%

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

      4. *-inverses 67.5%

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

      5. *-rgt-identity 67.5%

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

   7. Simplified 67.5%

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

      1. unpow2 67.5%

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

      2. associate-/l* 71.6%

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

   9. Applied egg-rr 71.6%

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

   if 1.25e172 < x

   1. Initial program 53.3%

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

      1. clear-num 53.3%

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

      2. inv-pow 53.3%

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

      3. associate-/l* 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. pow2 53.3%

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

      6. hypot-define 53.3%

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

      7. pow2 53.3%

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

   4. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{2}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 53.3%

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

      2. associate-*r/ 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x around inf 61.8%

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

      1. associate-*r/ 61.8%

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

   9. Simplified 61.8%

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

       1. clear-num 61.8%

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

       2. unpow2 61.8%

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

       3. *-commutative 61.8%

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

       4. times-frac 67.8%

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

   11. Applied egg-rr 67.8%

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

       1. *-commutative 67.8%

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

       2. clear-num 67.8%

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

       3. un-div-inv 67.9%

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

       4. div-inv 67.9%

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

       5. metadata-eval 67.9%

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

   13. Applied egg-rr 67.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+91}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 4.8e+91) (* y_m 0.5) (* z (* z (/ -0.5 y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 4.8e+91) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * (z * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 4.8d+91) then
        tmp = y_m * 0.5d0
    else
        tmp = z * (z * ((-0.5d0) / y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 4.8e+91) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * (z * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 4.8e+91:
		tmp = y_m * 0.5
	else:
		tmp = z * (z * (-0.5 / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 4.8e+91)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 4.8e+91)
		tmp = y_m * 0.5;
	else
		tmp = z * (z * (-0.5 / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 4.8e+91], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.79999999999999966e91

   1. Initial program 70.8%

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

   3. Taylor expanded in y around inf 38.3%

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

      1. *-commutative 38.3%

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

   5. Simplified 38.3%

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

   if 4.79999999999999966e91 < z

   1. Initial program 61.9%

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

   3. Taylor expanded in z around inf 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l/ 66.7%

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

   5. Simplified 66.7%

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

      1. associate-/l* 66.7%

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

      2. pow2 66.7%

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

      3. associate-*l* 72.6%

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

   7. Applied egg-rr 72.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+91}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 1.1e+93) (* y_m 0.5) (* z (/ (/ z -2.0) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.1e+93) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.1d+93) then
        tmp = y_m * 0.5d0
    else
        tmp = z * ((z / (-2.0d0)) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.1e+93) {
		tmp = y_m * 0.5;
	} else {
		tmp = z * ((z / -2.0) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1.1e+93:
		tmp = y_m * 0.5
	else:
		tmp = z * ((z / -2.0) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1.1e+93)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z * Float64(Float64(z / -2.0) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.1e+93)
		tmp = y_m * 0.5;
	else
		tmp = z * ((z / -2.0) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.1e+93], N[(y$95$m * 0.5), $MachinePrecision], N[(z * N[(N[(z / -2.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.10000000000000011e93

   1. Initial program 70.8%

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

   3. Taylor expanded in y around inf 38.3%

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

      1. *-commutative 38.3%

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

   5. Simplified 38.3%

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

   if 1.10000000000000011e93 < z

   1. Initial program 61.9%

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

   3. Taylor expanded in z around inf 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l/ 66.7%

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

   5. Simplified 66.7%

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

      1. associate-/l* 66.7%

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

      2. pow2 66.7%

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

      3. associate-*l* 72.6%

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

   7. Applied egg-rr 72.6%

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

      1. clear-num 72.6%

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

      2. un-div-inv 72.6%

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

      3. div-inv 72.6%

         \[\leadsto z \cdot \frac{z}{\color{blue}{y \cdot \frac{1}{-0.5}}} \]

      4. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. *-commutative 72.6%

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

       2. associate-/r* 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{-2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

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

Fortran

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

3. Taylor expanded in y around inf 34.4%

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

   1. *-commutative 34.4%

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

5. Simplified 34.4%

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

6. Final simplification 34.4%

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

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024078 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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