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

Percentage Accurate: 69.4% → 95.9%

Time: 10.9s

Alternatives: 9

Speedup: 0.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% 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: 95.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.6 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m - z \cdot z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + x \cdot \frac{x}{y\_m}\right) - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 6.6e-31)
    (* 0.5 (/ (fma x x (- (* y_m y_m) (* z z))) y_m))
    (* 0.5 (- (+ y_m (* x (/ x 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 <= 6.6e-31) {
		tmp = 0.5 * (fma(x, x, ((y_m * y_m) - (z * z))) / y_m);
	} else {
		tmp = 0.5 * ((y_m + (x * (x / 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 <= 6.6e-31)
		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(Float64(y_m + Float64(x * Float64(x / 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, 6.6e-31], 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[(N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.5999999999999998e-31

   1. Initial program 80.1%

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

      1. remove-double-neg 80.1%

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

      2. distribute-lft-neg-out 80.1%

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

      3. distribute-frac-neg2 80.1%

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

      4. distribute-frac-neg 80.1%

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

      5. neg-mul-1 80.1%

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

      6. distribute-lft-neg-out 80.1%

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

      7. *-commutative 80.1%

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

      8. distribute-lft-neg-in 80.1%

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

      9. times-frac 80.1%

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

      10. metadata-eval 80.1%

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

      11. metadata-eval 80.1%

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

      12. associate--l+ 80.1%

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

      13. fma-define 82.1%

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

   3. Simplified 82.1%

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

   if 6.5999999999999998e-31 < y

   1. Initial program 46.6%

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

      1. remove-double-neg 46.6%

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

      2. distribute-lft-neg-out 46.6%

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

      3. distribute-frac-neg2 46.6%

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

      4. distribute-frac-neg 46.6%

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

      5. neg-mul-1 46.6%

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

      6. distribute-lft-neg-out 46.6%

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

      7. *-commutative 46.6%

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

      8. distribute-lft-neg-in 46.6%

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

      9. times-frac 46.6%

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

      10. metadata-eval 46.6%

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

      11. metadata-eval 46.6%

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

      12. associate--l+ 46.6%

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

      13. fma-define 46.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in x around 0 80.0%

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

      1. pow2 80.0%

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

      2. associate-/l* 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. unpow2 90.1%

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

      2. associate-/l* 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := z \cdot \left(\frac{z}{y\_m} \cdot -0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-229}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-211}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* z (* (/ z y_m) -0.5))))
   (*
    y_s
    (if (<= x 2.45e-275)
      t_0
      (if (<= x 1.75e-229)
        (* y_m 0.5)
        (if (<= x 1.65e-211)
          t_0
          (if (<= x 3.6e-147)
            (* y_m 0.5)
            (if (<= x 5e+24) t_0 (* x (/ (* 0.5 x) 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 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 2.45e-275) {
		tmp = t_0;
	} else if (x <= 1.75e-229) {
		tmp = y_m * 0.5;
	} else if (x <= 1.65e-211) {
		tmp = t_0;
	} else if (x <= 3.6e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 5e+24) {
		tmp = t_0;
	} else {
		tmp = x * ((0.5 * x) / 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 = z * ((z / y_m) * (-0.5d0))
    if (x <= 2.45d-275) then
        tmp = t_0
    else if (x <= 1.75d-229) then
        tmp = y_m * 0.5d0
    else if (x <= 1.65d-211) then
        tmp = t_0
    else if (x <= 3.6d-147) then
        tmp = y_m * 0.5d0
    else if (x <= 5d+24) then
        tmp = t_0
    else
        tmp = x * ((0.5d0 * x) / 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 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 2.45e-275) {
		tmp = t_0;
	} else if (x <= 1.75e-229) {
		tmp = y_m * 0.5;
	} else if (x <= 1.65e-211) {
		tmp = t_0;
	} else if (x <= 3.6e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 5e+24) {
		tmp = t_0;
	} else {
		tmp = x * ((0.5 * x) / 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 = z * ((z / y_m) * -0.5)
	tmp = 0
	if x <= 2.45e-275:
		tmp = t_0
	elif x <= 1.75e-229:
		tmp = y_m * 0.5
	elif x <= 1.65e-211:
		tmp = t_0
	elif x <= 3.6e-147:
		tmp = y_m * 0.5
	elif x <= 5e+24:
		tmp = t_0
	else:
		tmp = x * ((0.5 * x) / 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(z * Float64(Float64(z / y_m) * -0.5))
	tmp = 0.0
	if (x <= 2.45e-275)
		tmp = t_0;
	elseif (x <= 1.75e-229)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.65e-211)
		tmp = t_0;
	elseif (x <= 3.6e-147)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 5e+24)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(0.5 * x) / 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 = z * ((z / y_m) * -0.5);
	tmp = 0.0;
	if (x <= 2.45e-275)
		tmp = t_0;
	elseif (x <= 1.75e-229)
		tmp = y_m * 0.5;
	elseif (x <= 1.65e-211)
		tmp = t_0;
	elseif (x <= 3.6e-147)
		tmp = y_m * 0.5;
	elseif (x <= 5e+24)
		tmp = t_0;
	else
		tmp = x * ((0.5 * x) / 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[(z * N[(N[(z / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 2.45e-275], t$95$0, If[LessEqual[x, 1.75e-229], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.65e-211], t$95$0, If[LessEqual[x, 3.6e-147], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 5e+24], t$95$0, N[(x * N[(N[(0.5 * x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 2.44999999999999991e-275 or 1.7500000000000002e-229 < x < 1.6500000000000001e-211 or 3.60000000000000012e-147 < x < 5.00000000000000045e24

   1. Initial program 75.6%

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

      1. clear-num 75.6%

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

      2. associate-/r/ 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-/r* 75.6%

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

      5. metadata-eval 75.6%

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

      6. add-sqr-sqrt 75.6%

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

      7. pow2 75.6%

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

      8. hypot-define 75.6%

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

      9. pow2 75.6%

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

   4. Applied egg-rr 75.6%

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

   5. Taylor expanded in z around inf 41.6%

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

      1. associate-*r/ 41.6%

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

   7. Simplified 41.6%

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

      1. associate-/l* 41.6%

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

      2. unpow2 41.6%

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

      3. associate-*l/ 43.2%

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

      4. associate-*r* 43.2%

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

   9. Applied egg-rr 43.2%

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

   if 2.44999999999999991e-275 < x < 1.7500000000000002e-229 or 1.6500000000000001e-211 < x < 3.60000000000000012e-147

   1. Initial program 46.5%

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

   3. Taylor expanded in y around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

   if 5.00000000000000045e24 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 74.2%

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

   5. Simplified 74.2%

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

      1. associate-/l* 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*l* 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-275}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-229}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-211}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := z \cdot \left(\frac{z}{y\_m} \cdot -0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.38 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-226}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-211}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y\_m}{x}}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* z (* (/ z y_m) -0.5))))
   (*
    y_s
    (if (<= x 1.38e-275)
      t_0
      (if (<= x 5.3e-226)
        (* y_m 0.5)
        (if (<= x 1.35e-211)
          t_0
          (if (<= x 3.5e-147)
            (* y_m 0.5)
            (if (<= x 1.2e+24) t_0 (/ (* 0.5 x) (/ y_m x))))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 1.38e-275) {
		tmp = t_0;
	} else if (x <= 5.3e-226) {
		tmp = y_m * 0.5;
	} else if (x <= 1.35e-211) {
		tmp = t_0;
	} else if (x <= 3.5e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 1.2e+24) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((z / y_m) * (-0.5d0))
    if (x <= 1.38d-275) then
        tmp = t_0
    else if (x <= 5.3d-226) then
        tmp = y_m * 0.5d0
    else if (x <= 1.35d-211) then
        tmp = t_0
    else if (x <= 3.5d-147) then
        tmp = y_m * 0.5d0
    else if (x <= 1.2d+24) then
        tmp = t_0
    else
        tmp = (0.5d0 * x) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 1.38e-275) {
		tmp = t_0;
	} else if (x <= 5.3e-226) {
		tmp = y_m * 0.5;
	} else if (x <= 1.35e-211) {
		tmp = t_0;
	} else if (x <= 3.5e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 1.2e+24) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = z * ((z / y_m) * -0.5)
	tmp = 0
	if x <= 1.38e-275:
		tmp = t_0
	elif x <= 5.3e-226:
		tmp = y_m * 0.5
	elif x <= 1.35e-211:
		tmp = t_0
	elif x <= 3.5e-147:
		tmp = y_m * 0.5
	elif x <= 1.2e+24:
		tmp = t_0
	else:
		tmp = (0.5 * x) / (y_m / x)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(z * Float64(Float64(z / y_m) * -0.5))
	tmp = 0.0
	if (x <= 1.38e-275)
		tmp = t_0;
	elseif (x <= 5.3e-226)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.35e-211)
		tmp = t_0;
	elseif (x <= 3.5e-147)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.2e+24)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * x) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = z * ((z / y_m) * -0.5);
	tmp = 0.0;
	if (x <= 1.38e-275)
		tmp = t_0;
	elseif (x <= 5.3e-226)
		tmp = y_m * 0.5;
	elseif (x <= 1.35e-211)
		tmp = t_0;
	elseif (x <= 3.5e-147)
		tmp = y_m * 0.5;
	elseif (x <= 1.2e+24)
		tmp = t_0;
	else
		tmp = (0.5 * x) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[(N[(z / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 1.38e-275], t$95$0, If[LessEqual[x, 5.3e-226], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.35e-211], t$95$0, If[LessEqual[x, 3.5e-147], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.2e+24], t$95$0, N[(N[(0.5 * x), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 1.37999999999999997e-275 or 5.3000000000000004e-226 < x < 1.35e-211 or 3.50000000000000004e-147 < x < 1.2e24

   1. Initial program 75.6%

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

      1. clear-num 75.6%

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

      2. associate-/r/ 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-/r* 75.6%

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

      5. metadata-eval 75.6%

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

      6. add-sqr-sqrt 75.6%

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

      7. pow2 75.6%

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

      8. hypot-define 75.6%

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

      9. pow2 75.6%

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

   4. Applied egg-rr 75.6%

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

   5. Taylor expanded in z around inf 41.6%

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

      1. associate-*r/ 41.6%

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

   7. Simplified 41.6%

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

      1. associate-/l* 41.6%

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

      2. unpow2 41.6%

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

      3. associate-*l/ 43.2%

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

      4. associate-*r* 43.2%

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

   9. Applied egg-rr 43.2%

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

   if 1.37999999999999997e-275 < x < 5.3000000000000004e-226 or 1.35e-211 < x < 3.50000000000000004e-147

   1. Initial program 46.5%

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

   3. Taylor expanded in y around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

   if 1.2e24 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 74.2%

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

   5. Simplified 74.2%

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

      1. associate-/l* 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*l* 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. associate-*r/ 74.2%

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

       2. *-commutative 74.2%

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

       3. associate-/l* 77.1%

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

       4. clear-num 77.1%

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

       5. un-div-inv 77.1%

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

   11. Applied egg-rr 77.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.38 \cdot 10^{-275}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-226}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-211}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := z \cdot \left(\frac{z}{y\_m} \cdot -0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-228}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot -0.5\right)}{y\_m}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-147}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y\_m}{x}}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* z (* (/ z y_m) -0.5))))
   (*
    y_s
    (if (<= x 2.45e-275)
      t_0
      (if (<= x 4.5e-228)
        (* y_m 0.5)
        (if (<= x 1.3e-211)
          (/ (* z (* z -0.5)) y_m)
          (if (<= x 6.8e-147)
            (* y_m 0.5)
            (if (<= x 4.6e+24) t_0 (/ (* 0.5 x) (/ y_m x))))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 2.45e-275) {
		tmp = t_0;
	} else if (x <= 4.5e-228) {
		tmp = y_m * 0.5;
	} else if (x <= 1.3e-211) {
		tmp = (z * (z * -0.5)) / y_m;
	} else if (x <= 6.8e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 4.6e+24) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((z / y_m) * (-0.5d0))
    if (x <= 2.45d-275) then
        tmp = t_0
    else if (x <= 4.5d-228) then
        tmp = y_m * 0.5d0
    else if (x <= 1.3d-211) then
        tmp = (z * (z * (-0.5d0))) / y_m
    else if (x <= 6.8d-147) then
        tmp = y_m * 0.5d0
    else if (x <= 4.6d+24) then
        tmp = t_0
    else
        tmp = (0.5d0 * x) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = z * ((z / y_m) * -0.5);
	double tmp;
	if (x <= 2.45e-275) {
		tmp = t_0;
	} else if (x <= 4.5e-228) {
		tmp = y_m * 0.5;
	} else if (x <= 1.3e-211) {
		tmp = (z * (z * -0.5)) / y_m;
	} else if (x <= 6.8e-147) {
		tmp = y_m * 0.5;
	} else if (x <= 4.6e+24) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = z * ((z / y_m) * -0.5)
	tmp = 0
	if x <= 2.45e-275:
		tmp = t_0
	elif x <= 4.5e-228:
		tmp = y_m * 0.5
	elif x <= 1.3e-211:
		tmp = (z * (z * -0.5)) / y_m
	elif x <= 6.8e-147:
		tmp = y_m * 0.5
	elif x <= 4.6e+24:
		tmp = t_0
	else:
		tmp = (0.5 * x) / (y_m / x)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(z * Float64(Float64(z / y_m) * -0.5))
	tmp = 0.0
	if (x <= 2.45e-275)
		tmp = t_0;
	elseif (x <= 4.5e-228)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 1.3e-211)
		tmp = Float64(Float64(z * Float64(z * -0.5)) / y_m);
	elseif (x <= 6.8e-147)
		tmp = Float64(y_m * 0.5);
	elseif (x <= 4.6e+24)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * x) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = z * ((z / y_m) * -0.5);
	tmp = 0.0;
	if (x <= 2.45e-275)
		tmp = t_0;
	elseif (x <= 4.5e-228)
		tmp = y_m * 0.5;
	elseif (x <= 1.3e-211)
		tmp = (z * (z * -0.5)) / y_m;
	elseif (x <= 6.8e-147)
		tmp = y_m * 0.5;
	elseif (x <= 4.6e+24)
		tmp = t_0;
	else
		tmp = (0.5 * x) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[(N[(z / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 2.45e-275], t$95$0, If[LessEqual[x, 4.5e-228], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 1.3e-211], N[(N[(z * N[(z * -0.5), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 6.8e-147], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[x, 4.6e+24], t$95$0, N[(N[(0.5 * x), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 2.44999999999999991e-275 or 6.79999999999999991e-147 < x < 4.5999999999999998e24

   1. Initial program 75.3%

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

      1. clear-num 75.3%

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

      2. associate-/r/ 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-/r* 75.2%

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

      5. metadata-eval 75.2%

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

      6. add-sqr-sqrt 75.2%

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

      7. pow2 75.2%

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

      8. hypot-define 75.2%

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

      9. pow2 75.2%

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

   4. Applied egg-rr 75.2%

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

   5. Taylor expanded in z around inf 40.1%

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

      1. associate-*r/ 40.1%

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

   7. Simplified 40.1%

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

      1. associate-/l* 40.1%

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

      2. unpow2 40.1%

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

      3. associate-*l/ 41.7%

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

      4. associate-*r* 41.7%

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

   9. Applied egg-rr 41.7%

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

   if 2.44999999999999991e-275 < x < 4.4999999999999999e-228 or 1.3e-211 < x < 6.79999999999999991e-147

   1. Initial program 46.5%

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

   3. Taylor expanded in y around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

   if 4.4999999999999999e-228 < x < 1.3e-211

   1. Initial program 84.5%

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

      1. clear-num 84.2%

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

      2. associate-/r/ 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-/r* 84.8%

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

      5. metadata-eval 84.8%

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

      6. add-sqr-sqrt 84.8%

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

      7. pow2 84.8%

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

      8. hypot-define 84.8%

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

      9. pow2 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in z around inf 83.4%

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

      1. associate-*r/ 83.4%

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

   7. Simplified 83.4%

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

      1. associate-/l* 83.4%

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

      2. unpow2 83.4%

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

      3. associate-*l/ 83.1%

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

      4. associate-*r* 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. associate-*r/ 83.1%

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

       2. associate-*l/ 83.4%

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

       3. *-commutative 83.4%

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

   11. Applied egg-rr 83.4%

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

   if 4.5999999999999998e24 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 74.2%

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

   5. Simplified 74.2%

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

      1. associate-/l* 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*l* 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. associate-*r/ 74.2%

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

       2. *-commutative 74.2%

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

       3. associate-/l* 77.1%

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

       4. clear-num 77.1%

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

       5. un-div-inv 77.1%

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

   11. Applied egg-rr 77.1%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-275}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-228}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot -0.5\right)}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + x \cdot \frac{x}{y\_m}\right) - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.35e+193)
    (* 0.5 (- (+ y_m (* x (/ x y_m))) (* z (/ z y_m))))
    (/ (* 0.5 x) (/ y_m x)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+193) {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z * (z / y_m)));
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.35d+193) then
        tmp = 0.5d0 * ((y_m + (x * (x / y_m))) - (z * (z / y_m)))
    else
        tmp = (0.5d0 * x) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+193) {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z * (z / y_m)));
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.35e+193:
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z * (z / y_m)))
	else:
		tmp = (0.5 * x) / (y_m / x)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.35e+193)
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x * Float64(x / y_m))) - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(0.5 * x) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.35e+193)
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z * (z / y_m)));
	else
		tmp = (0.5 * x) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.35e+193], N[(0.5 * N[(N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e193

   1. Initial program 73.4%

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

      1. remove-double-neg 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. distribute-frac-neg2 73.4%

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

      4. distribute-frac-neg 73.4%

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

      5. neg-mul-1 73.4%

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

      6. distribute-lft-neg-out 73.4%

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

      7. *-commutative 73.4%

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

      8. distribute-lft-neg-in 73.4%

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

      9. times-frac 73.4%

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

      10. metadata-eval 73.4%

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

      11. metadata-eval 73.4%

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

      12. associate--l+ 73.4%

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

      13. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in x around 0 87.3%

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

      1. pow2 87.3%

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

      2. associate-/l* 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. unpow2 91.0%

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

      2. associate-/l* 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 1.35e193 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

   5. Simplified 85.7%

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

      1. associate-/l* 85.7%

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

      2. unpow2 85.7%

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

      3. associate-*l* 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. associate-*r/ 88.9%

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

   9. Applied egg-rr 88.9%

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

       1. associate-*r/ 85.7%

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

       2. *-commutative 85.7%

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

       3. associate-/l* 88.9%

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

       4. clear-num 88.9%

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

       5. un-div-inv 88.9%

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

   11. Applied egg-rr 88.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + x \cdot \frac{x}{y\_m}\right) - \frac{z}{\frac{y\_m}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y\_m}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.45e+193)
    (* 0.5 (- (+ y_m (* x (/ x y_m))) (/ z (/ y_m z))))
    (/ (* 0.5 x) (/ y_m x)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.45e+193) {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.45d+193) then
        tmp = 0.5d0 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
    else
        tmp = (0.5d0 * x) / (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.45e+193) {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	} else {
		tmp = (0.5 * x) / (y_m / x);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.45e+193:
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
	else:
		tmp = (0.5 * x) / (y_m / x)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.45e+193)
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x * Float64(x / y_m))) - Float64(z / Float64(y_m / z))));
	else
		tmp = Float64(Float64(0.5 * x) / Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.45e+193)
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	else
		tmp = (0.5 * x) / (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.45e+193], N[(0.5 * N[(N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000007e193

   1. Initial program 73.4%

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

      1. remove-double-neg 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. distribute-frac-neg2 73.4%

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

      4. distribute-frac-neg 73.4%

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

      5. neg-mul-1 73.4%

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

      6. distribute-lft-neg-out 73.4%

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

      7. *-commutative 73.4%

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

      8. distribute-lft-neg-in 73.4%

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

      9. times-frac 73.4%

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

      10. metadata-eval 73.4%

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

      11. metadata-eval 73.4%

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

      12. associate--l+ 73.4%

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

      13. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in x around 0 87.3%

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

      1. pow2 87.3%

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

      2. associate-/l* 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. unpow2 91.0%

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

      2. associate-/l* 94.3%

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

   9. Applied egg-rr 94.3%

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

       1. clear-num 94.2%

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

       2. un-div-inv 94.3%

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

   11. Applied egg-rr 94.3%

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

   if 1.45000000000000007e193 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

   5. Simplified 85.7%

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

      1. associate-/l* 85.7%

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

      2. unpow2 85.7%

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

      3. associate-*l* 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. associate-*r/ 88.9%

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

   9. Applied egg-rr 88.9%

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

       1. associate-*r/ 85.7%

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

       2. *-commutative 85.7%

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

       3. associate-/l* 88.9%

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

       4. clear-num 88.9%

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

       5. un-div-inv 88.9%

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

   11. Applied egg-rr 88.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1.35e+25) (* y_m 0.5) (* x (* x (/ 0.5 y_m))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+25) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (0.5 / y_m));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.35d+25) then
        tmp = y_m * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y_m))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+25) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * (x * (0.5 / y_m));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.35e+25:
		tmp = y_m * 0.5
	else:
		tmp = x * (x * (0.5 / y_m))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.35e+25)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.35e+25)
		tmp = y_m * 0.5;
	else
		tmp = x * (x * (0.5 / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.35e+25], N[(y$95$m * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e25

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf 33.4%

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

      1. *-commutative 33.4%

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

   5. Simplified 33.4%

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

   if 1.35e25 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 74.2%

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

   5. Simplified 74.2%

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

      1. associate-/l* 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*l* 77.1%

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

   7. Applied egg-rr 77.1%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+25}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1.1e+25) (* y_m 0.5) (* x (/ (* 0.5 x) y_m)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.1e+25) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * ((0.5 * x) / y_m);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.1d+25) then
        tmp = y_m * 0.5d0
    else
        tmp = x * ((0.5d0 * x) / y_m)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.1e+25) {
		tmp = y_m * 0.5;
	} else {
		tmp = x * ((0.5 * x) / y_m);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.1e+25:
		tmp = y_m * 0.5
	else:
		tmp = x * ((0.5 * x) / y_m)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.1e+25)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(x * Float64(Float64(0.5 * x) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.1e+25)
		tmp = y_m * 0.5;
	else
		tmp = x * ((0.5 * x) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.1e+25], N[(y$95$m * 0.5), $MachinePrecision], N[(x * N[(N[(0.5 * x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e25

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf 33.4%

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

      1. *-commutative 33.4%

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

   5. Simplified 33.4%

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

   if 1.1e25 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l/ 74.2%

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

   5. Simplified 74.2%

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

      1. associate-/l* 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*l* 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.4% 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 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in y around inf 28.4%

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

   1. *-commutative 28.4%

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

5. Simplified 28.4%

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

6. Final simplification 28.4%

   \[\leadsto y \cdot 0.5 \]
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 2024039 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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