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

Percentage Accurate: 68.5% → 90.1%

Time: 14.7s

Alternatives: 11

Speedup: 0.8×

• 41.0% 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: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+173} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z + y, y - z, x \cdot x\right)}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e+173) (not (<= y 3.5e+162)))
   (* (/ (+ z y) (* y 2.0)) (- y z))
   (/ (fma (+ z y) (- y z) (* x x)) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+173) || !(y <= 3.5e+162)) {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	} else {
		tmp = fma((z + y), (y - z), (x * x)) / (y * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e+173) || !(y <= 3.5e+162))
		tmp = Float64(Float64(Float64(z + y) / Float64(y * 2.0)) * Float64(y - z));
	else
		tmp = Float64(fma(Float64(z + y), Float64(y - z), Float64(x * x)) / Float64(y * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e+173], N[Not[LessEqual[y, 3.5e+162]], $MachinePrecision]], N[(N[(N[(z + y), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] * N[(y - z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e173 or 3.50000000000000018e162 < y

   1. Initial program 5.2%

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

      1. associate--l+ 5.2%

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

      2. +-commutative 5.2%

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

      3. sqr-neg 5.2%

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

      4. difference-of-squares 12.3%

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

      5. sub-neg 12.3%

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

      6. remove-double-neg 12.3%

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

      7. *-commutative 12.3%

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

      8. fma-def 12.4%

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

      9. sub-neg 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in x around 0 12.3%

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

      1. associate-/l* 93.3%

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

      2. associate-/r/ 93.4%

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

   6. Applied egg-rr 93.4%

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

   if -4.2e173 < y < 3.50000000000000018e162

   1. Initial program 87.2%

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

      1. associate--l+ 87.2%

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

      2. +-commutative 87.2%

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

      3. sqr-neg 87.2%

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

      4. difference-of-squares 87.7%

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

      5. sub-neg 87.7%

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

      6. remove-double-neg 87.7%

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

      7. *-commutative 87.7%

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

      8. fma-def 89.8%

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

      9. sub-neg 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+173} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z + y, y - z, x \cdot x\right)}{y \cdot 2}\\ \end{array} \]

Alternative 2: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+129}:\\ \;\;\;\;t_0 \cdot \left(\frac{0.5}{y} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot (hypot x y) z)))
   (if (<= (* z z) 2e+129)
     (* t_0 (* (/ 0.5 y) t_0))
     (* (/ (+ z y) (* y 2.0)) (- y z)))))

C

double code(double x, double y, double z) {
	double t_0 = hypot(hypot(x, y), z);
	double tmp;
	if ((z * z) <= 2e+129) {
		tmp = t_0 * ((0.5 / y) * t_0);
	} else {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(Math.hypot(x, y), z);
	double tmp;
	if ((z * z) <= 2e+129) {
		tmp = t_0 * ((0.5 / y) * t_0);
	} else {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.hypot(math.hypot(x, y), z)
	tmp = 0
	if (z * z) <= 2e+129:
		tmp = t_0 * ((0.5 / y) * t_0)
	else:
		tmp = ((z + y) / (y * 2.0)) * (y - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = hypot(hypot(x, y), z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+129)
		tmp = Float64(t_0 * Float64(Float64(0.5 / y) * t_0));
	else
		tmp = Float64(Float64(Float64(z + y) / Float64(y * 2.0)) * Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = hypot(hypot(x, y), z);
	tmp = 0.0;
	if ((z * z) <= 2e+129)
		tmp = t_0 * ((0.5 / y) * t_0);
	else
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] ^ 2 + z ^ 2], $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e+129], N[(t$95$0 * N[(N[(0.5 / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e129

   1. Initial program 77.2%

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

      1. div-inv 77.1%

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

      2. *-commutative 77.1%

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

      3. *-commutative 77.1%

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

      4. associate-/r* 77.1%

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

      5. metadata-eval 77.1%

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

      6. add-sqr-sqrt 77.1%

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

      7. pow2 77.1%

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

      8. hypot-def 77.1%

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

      9. pow2 77.1%

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

   3. Applied egg-rr 77.1%

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

      1. clear-num 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-*l/ 77.2%

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

      4. div-inv 77.2%

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

      5. metadata-eval 77.2%

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

      6. associate-/l* 77.1%

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

      7. div-inv 77.1%

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

      8. associate-/r* 77.1%

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

      9. metadata-eval 77.1%

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

      10. metadata-eval 77.1%

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

      11. div-inv 77.1%

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

      12. clear-num 77.1%

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

   5. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{y}}{\frac{1}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 77.1%

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

      2. add-sqr-sqrt 70.0%

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

      3. associate-*r* 70.1%

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

   7. Applied egg-rr 92.1%

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

   if 2e129 < (*.f64 z z)

   1. Initial program 56.3%

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

      1. associate--l+ 56.3%

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

      2. +-commutative 56.3%

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

      3. sqr-neg 56.3%

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

      4. difference-of-squares 61.3%

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

      5. sub-neg 61.3%

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

      6. remove-double-neg 61.3%

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

      7. *-commutative 61.3%

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

      8. fma-def 65.5%

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

      9. sub-neg 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around 0 63.4%

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

      1. associate-/l* 85.2%

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

      2. associate-/r/ 85.3%

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

   6. Applied egg-rr 85.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right) \cdot \left(\frac{0.5}{y} \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \end{array} \]

Alternative 3: 87.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.65e+36) (not (<= y 3.5e+162)))
   (* (/ (+ z y) (* y 2.0)) (- y z))
   (/ (- (fma x x (* y y)) (* z z)) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.65e+36) || !(y <= 3.5e+162)) {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	} else {
		tmp = (fma(x, x, (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.65e+36) || !(y <= 3.5e+162))
		tmp = Float64(Float64(Float64(z + y) / Float64(y * 2.0)) * Float64(y - z));
	else
		tmp = Float64(Float64(fma(x, x, Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.65e+36], N[Not[LessEqual[y, 3.5e+162]], $MachinePrecision]], N[(N[(N[(z + y), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.65e36 or 3.50000000000000018e162 < y

   1. Initial program 30.9%

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

      1. associate--l+ 30.9%

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

      2. +-commutative 30.9%

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

      3. sqr-neg 30.9%

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

      4. difference-of-squares 36.6%

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

      5. sub-neg 36.6%

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

      6. remove-double-neg 36.6%

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

      7. *-commutative 36.6%

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

      8. fma-def 37.9%

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

      9. sub-neg 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in x around 0 35.5%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if -2.65e36 < y < 3.50000000000000018e162

   1. Initial program 88.9%

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

      1. div-sub 82.4%

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

      2. sqr-neg 82.4%

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

      3. div-sub 88.9%

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

      4. fma-def 88.9%

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

      5. sqr-neg 88.9%

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

   3. Simplified 88.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 4: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ t_1 := x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) (* (/ 0.5 y) (+ z y)))) (t_1 (* x (* (/ 0.5 y) x))))
   (if (<= x 1.85e+57)
     t_0
     (if (<= x 3.7e+86)
       (/ (* x (* 0.5 x)) y)
       (if (<= x 6.8e+111)
         t_0
         (if (<= x 9e+119)
           t_1
           (if (<= x 5.3e+129)
             t_0
             (if (<= x 5.5e+182)
               t_1
               (if (<= x 1.05e+195) t_0 (/ x (* 2.0 (/ y x))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * ((0.5 / y) * (z + y));
	double t_1 = x * ((0.5 / y) * x);
	double tmp;
	if (x <= 1.85e+57) {
		tmp = t_0;
	} else if (x <= 3.7e+86) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 6.8e+111) {
		tmp = t_0;
	} else if (x <= 9e+119) {
		tmp = t_1;
	} else if (x <= 5.3e+129) {
		tmp = t_0;
	} else if (x <= 5.5e+182) {
		tmp = t_1;
	} else if (x <= 1.05e+195) {
		tmp = t_0;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - z) * ((0.5d0 / y) * (z + y))
    t_1 = x * ((0.5d0 / y) * x)
    if (x <= 1.85d+57) then
        tmp = t_0
    else if (x <= 3.7d+86) then
        tmp = (x * (0.5d0 * x)) / y
    else if (x <= 6.8d+111) then
        tmp = t_0
    else if (x <= 9d+119) then
        tmp = t_1
    else if (x <= 5.3d+129) then
        tmp = t_0
    else if (x <= 5.5d+182) then
        tmp = t_1
    else if (x <= 1.05d+195) then
        tmp = t_0
    else
        tmp = x / (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) * ((0.5 / y) * (z + y));
	double t_1 = x * ((0.5 / y) * x);
	double tmp;
	if (x <= 1.85e+57) {
		tmp = t_0;
	} else if (x <= 3.7e+86) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 6.8e+111) {
		tmp = t_0;
	} else if (x <= 9e+119) {
		tmp = t_1;
	} else if (x <= 5.3e+129) {
		tmp = t_0;
	} else if (x <= 5.5e+182) {
		tmp = t_1;
	} else if (x <= 1.05e+195) {
		tmp = t_0;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) * ((0.5 / y) * (z + y))
	t_1 = x * ((0.5 / y) * x)
	tmp = 0
	if x <= 1.85e+57:
		tmp = t_0
	elif x <= 3.7e+86:
		tmp = (x * (0.5 * x)) / y
	elif x <= 6.8e+111:
		tmp = t_0
	elif x <= 9e+119:
		tmp = t_1
	elif x <= 5.3e+129:
		tmp = t_0
	elif x <= 5.5e+182:
		tmp = t_1
	elif x <= 1.05e+195:
		tmp = t_0
	else:
		tmp = x / (2.0 * (y / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * Float64(Float64(0.5 / y) * Float64(z + y)))
	t_1 = Float64(x * Float64(Float64(0.5 / y) * x))
	tmp = 0.0
	if (x <= 1.85e+57)
		tmp = t_0;
	elseif (x <= 3.7e+86)
		tmp = Float64(Float64(x * Float64(0.5 * x)) / y);
	elseif (x <= 6.8e+111)
		tmp = t_0;
	elseif (x <= 9e+119)
		tmp = t_1;
	elseif (x <= 5.3e+129)
		tmp = t_0;
	elseif (x <= 5.5e+182)
		tmp = t_1;
	elseif (x <= 1.05e+195)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) * ((0.5 / y) * (z + y));
	t_1 = x * ((0.5 / y) * x);
	tmp = 0.0;
	if (x <= 1.85e+57)
		tmp = t_0;
	elseif (x <= 3.7e+86)
		tmp = (x * (0.5 * x)) / y;
	elseif (x <= 6.8e+111)
		tmp = t_0;
	elseif (x <= 9e+119)
		tmp = t_1;
	elseif (x <= 5.3e+129)
		tmp = t_0;
	elseif (x <= 5.5e+182)
		tmp = t_1;
	elseif (x <= 1.05e+195)
		tmp = t_0;
	else
		tmp = x / (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * N[(N[(0.5 / y), $MachinePrecision] * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e+57], t$95$0, If[LessEqual[x, 3.7e+86], N[(N[(x * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.8e+111], t$95$0, If[LessEqual[x, 9e+119], t$95$1, If[LessEqual[x, 5.3e+129], t$95$0, If[LessEqual[x, 5.5e+182], t$95$1, If[LessEqual[x, 1.05e+195], t$95$0, N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.85000000000000003e57 or 3.69999999999999992e86 < x < 6.8000000000000003e111 or 9.00000000000000039e119 < x < 5.2999999999999999e129 or 5.49999999999999977e182 < x < 1.05000000000000005e195

   1. Initial program 69.6%

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

      1. associate--l+ 69.6%

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

      2. +-commutative 69.6%

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

      3. sqr-neg 69.6%

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

      4. difference-of-squares 71.9%

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

      5. sub-neg 71.9%

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

      6. remove-double-neg 71.9%

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

      7. *-commutative 71.9%

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

      8. fma-def 72.9%

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

      9. sub-neg 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 54.1%

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

      1. div-inv 54.0%

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

      2. *-commutative 54.0%

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

      3. associate-*l* 76.0%

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

      4. metadata-eval 76.0%

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

      5. div-inv 76.0%

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

      6. clear-num 76.0%

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

   6. Applied egg-rr 76.0%

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

   if 1.85000000000000003e57 < x < 3.69999999999999992e86

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.0%

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

      1. div-inv 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l* 75.0%

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

      4. metadata-eval 75.0%

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

      5. div-inv 75.0%

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

      6. clear-num 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*r/ 75.0%

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

      3. associate-*l/ 75.0%

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

      4. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

   if 6.8000000000000003e111 < x < 9.00000000000000039e119 or 5.2999999999999999e129 < x < 5.49999999999999977e182

   1. Initial program 68.7%

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

   2. Taylor expanded in x around inf 61.5%

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

      1. div-inv 61.5%

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

      2. unpow2 61.5%

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

      3. associate-*l* 67.9%

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

      4. metadata-eval 67.9%

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

      5. div-inv 67.9%

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

      6. clear-num 67.9%

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

   4. Applied egg-rr 67.9%

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

   if 1.05000000000000005e195 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 84.6%

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

      4. metadata-eval 84.6%

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

      5. div-inv 84.6%

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

      6. clear-num 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. div-inv 84.6%

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

      2. associate-*r* 84.6%

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

      3. *-commutative 84.6%

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

      4. associate-/r/ 84.7%

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

      5. div-inv 84.8%

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

      6. *-un-lft-identity 84.8%

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

      7. *-commutative 84.8%

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

      8. times-frac 84.8%

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

      9. metadata-eval 84.8%

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

   6. Applied egg-rr 84.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+57}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+195}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 5: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ t_1 := \left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ t_2 := x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ z y) (* y 2.0)) (- y z)))
        (t_1 (* (- y z) (* (/ 0.5 y) (+ z y))))
        (t_2 (* x (* (/ 0.5 y) x))))
   (if (<= x 1.95e+57)
     t_0
     (if (<= x 1.9e+86)
       (/ (* x (* 0.5 x)) y)
       (if (<= x 4.5e+111)
         t_1
         (if (<= x 2.8e+119)
           t_2
           (if (<= x 4.4e+130)
             t_1
             (if (<= x 2.35e+182)
               t_2
               (if (<= x 1e+195) t_0 (/ x (* 2.0 (/ y x))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = ((z + y) / (y * 2.0)) * (y - z);
	double t_1 = (y - z) * ((0.5 / y) * (z + y));
	double t_2 = x * ((0.5 / y) * x);
	double tmp;
	if (x <= 1.95e+57) {
		tmp = t_0;
	} else if (x <= 1.9e+86) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 4.5e+111) {
		tmp = t_1;
	} else if (x <= 2.8e+119) {
		tmp = t_2;
	} else if (x <= 4.4e+130) {
		tmp = t_1;
	} else if (x <= 2.35e+182) {
		tmp = t_2;
	} else if (x <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((z + y) / (y * 2.0d0)) * (y - z)
    t_1 = (y - z) * ((0.5d0 / y) * (z + y))
    t_2 = x * ((0.5d0 / y) * x)
    if (x <= 1.95d+57) then
        tmp = t_0
    else if (x <= 1.9d+86) then
        tmp = (x * (0.5d0 * x)) / y
    else if (x <= 4.5d+111) then
        tmp = t_1
    else if (x <= 2.8d+119) then
        tmp = t_2
    else if (x <= 4.4d+130) then
        tmp = t_1
    else if (x <= 2.35d+182) then
        tmp = t_2
    else if (x <= 1d+195) then
        tmp = t_0
    else
        tmp = x / (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z + y) / (y * 2.0)) * (y - z);
	double t_1 = (y - z) * ((0.5 / y) * (z + y));
	double t_2 = x * ((0.5 / y) * x);
	double tmp;
	if (x <= 1.95e+57) {
		tmp = t_0;
	} else if (x <= 1.9e+86) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 4.5e+111) {
		tmp = t_1;
	} else if (x <= 2.8e+119) {
		tmp = t_2;
	} else if (x <= 4.4e+130) {
		tmp = t_1;
	} else if (x <= 2.35e+182) {
		tmp = t_2;
	} else if (x <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z + y) / (y * 2.0)) * (y - z)
	t_1 = (y - z) * ((0.5 / y) * (z + y))
	t_2 = x * ((0.5 / y) * x)
	tmp = 0
	if x <= 1.95e+57:
		tmp = t_0
	elif x <= 1.9e+86:
		tmp = (x * (0.5 * x)) / y
	elif x <= 4.5e+111:
		tmp = t_1
	elif x <= 2.8e+119:
		tmp = t_2
	elif x <= 4.4e+130:
		tmp = t_1
	elif x <= 2.35e+182:
		tmp = t_2
	elif x <= 1e+195:
		tmp = t_0
	else:
		tmp = x / (2.0 * (y / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z + y) / Float64(y * 2.0)) * Float64(y - z))
	t_1 = Float64(Float64(y - z) * Float64(Float64(0.5 / y) * Float64(z + y)))
	t_2 = Float64(x * Float64(Float64(0.5 / y) * x))
	tmp = 0.0
	if (x <= 1.95e+57)
		tmp = t_0;
	elseif (x <= 1.9e+86)
		tmp = Float64(Float64(x * Float64(0.5 * x)) / y);
	elseif (x <= 4.5e+111)
		tmp = t_1;
	elseif (x <= 2.8e+119)
		tmp = t_2;
	elseif (x <= 4.4e+130)
		tmp = t_1;
	elseif (x <= 2.35e+182)
		tmp = t_2;
	elseif (x <= 1e+195)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z + y) / (y * 2.0)) * (y - z);
	t_1 = (y - z) * ((0.5 / y) * (z + y));
	t_2 = x * ((0.5 / y) * x);
	tmp = 0.0;
	if (x <= 1.95e+57)
		tmp = t_0;
	elseif (x <= 1.9e+86)
		tmp = (x * (0.5 * x)) / y;
	elseif (x <= 4.5e+111)
		tmp = t_1;
	elseif (x <= 2.8e+119)
		tmp = t_2;
	elseif (x <= 4.4e+130)
		tmp = t_1;
	elseif (x <= 2.35e+182)
		tmp = t_2;
	elseif (x <= 1e+195)
		tmp = t_0;
	else
		tmp = x / (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z + y), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(0.5 / y), $MachinePrecision] * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.95e+57], t$95$0, If[LessEqual[x, 1.9e+86], N[(N[(x * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4.5e+111], t$95$1, If[LessEqual[x, 2.8e+119], t$95$2, If[LessEqual[x, 4.4e+130], t$95$1, If[LessEqual[x, 2.35e+182], t$95$2, If[LessEqual[x, 1e+195], t$95$0, N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.94999999999999984e57 or 2.34999999999999992e182 < x < 9.99999999999999977e194

   1. Initial program 69.9%

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

      1. associate--l+ 69.9%

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

      2. +-commutative 69.9%

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

      3. sqr-neg 69.9%

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

      4. difference-of-squares 71.7%

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

      5. sub-neg 71.7%

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

      6. remove-double-neg 71.7%

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

      7. *-commutative 71.7%

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

      8. fma-def 72.8%

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

      9. sub-neg 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in x around 0 53.7%

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

      1. associate-/l* 75.6%

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

      2. associate-/r/ 75.6%

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

   6. Applied egg-rr 75.6%

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

   if 1.94999999999999984e57 < x < 1.89999999999999989e86

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.0%

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

      1. div-inv 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l* 75.0%

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

      4. metadata-eval 75.0%

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

      5. div-inv 75.0%

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

      6. clear-num 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*r/ 75.0%

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

      3. associate-*l/ 75.0%

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

      4. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

   if 1.89999999999999989e86 < x < 4.50000000000000001e111 or 2.80000000000000013e119 < x < 4.39999999999999987e130

   1. Initial program 62.5%

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

      1. associate--l+ 62.5%

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

      2. +-commutative 62.5%

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

      3. sqr-neg 62.5%

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

      4. difference-of-squares 77.4%

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

      5. sub-neg 77.4%

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

      6. remove-double-neg 77.4%

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

      7. *-commutative 77.4%

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

      8. fma-def 77.4%

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

      9. sub-neg 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 65.2%

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

      1. div-inv 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-*l* 87.8%

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

      4. metadata-eval 87.8%

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

      5. div-inv 87.8%

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

      6. clear-num 87.8%

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

   6. Applied egg-rr 87.8%

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

   if 4.50000000000000001e111 < x < 2.80000000000000013e119 or 4.39999999999999987e130 < x < 2.34999999999999992e182

   1. Initial program 68.7%

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

   2. Taylor expanded in x around inf 61.5%

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

      1. div-inv 61.5%

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

      2. unpow2 61.5%

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

      3. associate-*l* 67.9%

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

      4. metadata-eval 67.9%

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

      5. div-inv 67.9%

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

      6. clear-num 67.9%

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

   4. Applied egg-rr 67.9%

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

   if 9.99999999999999977e194 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 84.6%

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

      4. metadata-eval 84.6%

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

      5. div-inv 84.6%

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

      6. clear-num 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. div-inv 84.6%

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

      2. associate-*r* 84.6%

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

      3. *-commutative 84.6%

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

      4. associate-/r/ 84.7%

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

      5. div-inv 84.8%

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

      6. *-un-lft-identity 84.8%

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

      7. *-commutative 84.8%

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

      8. times-frac 84.8%

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

      9. metadata-eval 84.8%

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

   6. Applied egg-rr 84.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+57}:\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+111}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;\left(y - z\right) \cdot \left(\frac{0.5}{y} \cdot \left(z + y\right)\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 6: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.22 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.22e+36) (not (<= y 3.5e+162)))
   (* (/ (+ z y) (* y 2.0)) (- y z))
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.22e+36) || !(y <= 3.5e+162)) {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.22d+36)) .or. (.not. (y <= 3.5d+162))) then
        tmp = ((z + y) / (y * 2.0d0)) * (y - z)
    else
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.22e+36) || !(y <= 3.5e+162)) {
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.22e+36) or not (y <= 3.5e+162):
		tmp = ((z + y) / (y * 2.0)) * (y - z)
	else:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.22e+36) || !(y <= 3.5e+162))
		tmp = Float64(Float64(Float64(z + y) / Float64(y * 2.0)) * Float64(y - z));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.22e+36) || ~((y <= 3.5e+162)))
		tmp = ((z + y) / (y * 2.0)) * (y - z);
	else
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.22e+36], N[Not[LessEqual[y, 3.5e+162]], $MachinePrecision]], N[(N[(N[(z + y), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.22e36 or 3.50000000000000018e162 < y

   1. Initial program 30.9%

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

      1. associate--l+ 30.9%

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

      2. +-commutative 30.9%

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

      3. sqr-neg 30.9%

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

      4. difference-of-squares 36.6%

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

      5. sub-neg 36.6%

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

      6. remove-double-neg 36.6%

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

      7. *-commutative 36.6%

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

      8. fma-def 37.9%

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

      9. sub-neg 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in x around 0 35.5%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if -2.22e36 < y < 3.50000000000000018e162

   1. Initial program 88.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.22 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{z + y}{y \cdot 2} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 7: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+19} \lor \neg \left(x \leq 5.5 \cdot 10^{+182}\right) \land x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 1.6e+19) (and (not (<= x 5.5e+182)) (<= x 1e+195)))
   (* 0.5 y)
   (* x (* (/ 0.5 y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 1.6e+19) || (!(x <= 5.5e+182) && (x <= 1e+195))) {
		tmp = 0.5 * y;
	} else {
		tmp = x * ((0.5 / y) * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= 1.6d+19) .or. (.not. (x <= 5.5d+182)) .and. (x <= 1d+195)) then
        tmp = 0.5d0 * y
    else
        tmp = x * ((0.5d0 / y) * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 1.6e+19) || (!(x <= 5.5e+182) && (x <= 1e+195))) {
		tmp = 0.5 * y;
	} else {
		tmp = x * ((0.5 / y) * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 1.6e+19) or (not (x <= 5.5e+182) and (x <= 1e+195)):
		tmp = 0.5 * y
	else:
		tmp = x * ((0.5 / y) * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 1.6e+19) || (!(x <= 5.5e+182) && (x <= 1e+195)))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(Float64(0.5 / y) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= 1.6e+19) || (~((x <= 5.5e+182)) && (x <= 1e+195)))
		tmp = 0.5 * y;
	else
		tmp = x * ((0.5 / y) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 1.6e+19], And[N[Not[LessEqual[x, 5.5e+182]], $MachinePrecision], LessEqual[x, 1e+195]]], N[(0.5 * y), $MachinePrecision], N[(x * N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e19 or 5.49999999999999977e182 < x < 9.99999999999999977e194

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 41.3%

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

   if 1.6e19 < x < 5.49999999999999977e182 or 9.99999999999999977e194 < x

   1. Initial program 69.4%

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

   2. Taylor expanded in x around inf 56.7%

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

      1. div-inv 56.7%

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

      2. unpow2 56.7%

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

      3. associate-*l* 65.7%

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

      4. metadata-eval 65.7%

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

      5. div-inv 65.7%

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

      6. clear-num 65.7%

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

   4. Applied egg-rr 65.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+19} \lor \neg \left(x \leq 5.5 \cdot 10^{+182}\right) \land x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \end{array} \]

Alternative 8: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.4e+18)
   (* 0.5 y)
   (if (<= x 5.5e+182)
     (* x (* (/ 0.5 y) x))
     (if (<= x 1e+195) (* 0.5 y) (* x (/ 0.5 (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.4e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = x * ((0.5 / y) * x);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (0.5 / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 8.4d+18) then
        tmp = 0.5d0 * y
    else if (x <= 5.5d+182) then
        tmp = x * ((0.5d0 / y) * x)
    else if (x <= 1d+195) then
        tmp = 0.5d0 * y
    else
        tmp = x * (0.5d0 / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.4e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = x * ((0.5 / y) * x);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (0.5 / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 8.4e+18:
		tmp = 0.5 * y
	elif x <= 5.5e+182:
		tmp = x * ((0.5 / y) * x)
	elif x <= 1e+195:
		tmp = 0.5 * y
	else:
		tmp = x * (0.5 / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.4e+18)
		tmp = Float64(0.5 * y);
	elseif (x <= 5.5e+182)
		tmp = Float64(x * Float64(Float64(0.5 / y) * x));
	elseif (x <= 1e+195)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(0.5 / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 8.4e+18)
		tmp = 0.5 * y;
	elseif (x <= 5.5e+182)
		tmp = x * ((0.5 / y) * x);
	elseif (x <= 1e+195)
		tmp = 0.5 * y;
	else
		tmp = x * (0.5 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.4e+18], N[(0.5 * y), $MachinePrecision], If[LessEqual[x, 5.5e+182], N[(x * N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+195], N[(0.5 * y), $MachinePrecision], N[(x * N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.4e18 or 5.49999999999999977e182 < x < 9.99999999999999977e194

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 41.3%

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

   if 8.4e18 < x < 5.49999999999999977e182

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf 48.4%

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

      1. div-inv 48.5%

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

      2. unpow2 48.5%

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

      3. associate-*l* 51.2%

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

      4. metadata-eval 51.2%

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

      5. div-inv 51.2%

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

      6. clear-num 51.2%

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

   4. Applied egg-rr 51.2%

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

   if 9.99999999999999977e194 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 84.6%

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

      4. metadata-eval 84.6%

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

      5. div-inv 84.6%

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

      6. clear-num 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. associate-*r/ 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-/l* 84.7%

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

   6. Applied egg-rr 84.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(\frac{0.5}{y} \cdot x\right)\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{y}{x}}\\ \end{array} \]

Alternative 9: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.1 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.1e+18)
   (* 0.5 y)
   (if (<= x 5.5e+182)
     (* (/ x y) (/ x 2.0))
     (if (<= x 1e+195) (* 0.5 y) (* x (/ 0.5 (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.1e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = (x / y) * (x / 2.0);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (0.5 / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9.1d+18) then
        tmp = 0.5d0 * y
    else if (x <= 5.5d+182) then
        tmp = (x / y) * (x / 2.0d0)
    else if (x <= 1d+195) then
        tmp = 0.5d0 * y
    else
        tmp = x * (0.5d0 / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.1e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = (x / y) * (x / 2.0);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (0.5 / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.1e+18:
		tmp = 0.5 * y
	elif x <= 5.5e+182:
		tmp = (x / y) * (x / 2.0)
	elif x <= 1e+195:
		tmp = 0.5 * y
	else:
		tmp = x * (0.5 / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.1e+18)
		tmp = Float64(0.5 * y);
	elseif (x <= 5.5e+182)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	elseif (x <= 1e+195)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(0.5 / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.1e+18)
		tmp = 0.5 * y;
	elseif (x <= 5.5e+182)
		tmp = (x / y) * (x / 2.0);
	elseif (x <= 1e+195)
		tmp = 0.5 * y;
	else
		tmp = x * (0.5 / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.1e+18], N[(0.5 * y), $MachinePrecision], If[LessEqual[x, 5.5e+182], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+195], N[(0.5 * y), $MachinePrecision], N[(x * N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.1e18 or 5.49999999999999977e182 < x < 9.99999999999999977e194

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 41.3%

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

   if 9.1e18 < x < 5.49999999999999977e182

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf 48.4%

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

      1. unpow2 48.4%

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

      2. times-frac 51.3%

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

   4. Applied egg-rr 51.3%

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

   if 9.99999999999999977e194 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 84.6%

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

      4. metadata-eval 84.6%

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

      5. div-inv 84.6%

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

      6. clear-num 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. associate-*r/ 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-/l* 84.7%

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

   6. Applied egg-rr 84.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.1 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{y}{x}}\\ \end{array} \]

Alternative 10: 43.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.5e+18)
   (* 0.5 y)
   (if (<= x 5.5e+182)
     (* (/ x y) (/ x 2.0))
     (if (<= x 1e+195) (* 0.5 y) (/ x (* 2.0 (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.5e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = (x / y) * (x / 2.0);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.5d+18) then
        tmp = 0.5d0 * y
    else if (x <= 5.5d+182) then
        tmp = (x / y) * (x / 2.0d0)
    else if (x <= 1d+195) then
        tmp = 0.5d0 * y
    else
        tmp = x / (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.5e+18) {
		tmp = 0.5 * y;
	} else if (x <= 5.5e+182) {
		tmp = (x / y) * (x / 2.0);
	} else if (x <= 1e+195) {
		tmp = 0.5 * y;
	} else {
		tmp = x / (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.5e+18:
		tmp = 0.5 * y
	elif x <= 5.5e+182:
		tmp = (x / y) * (x / 2.0)
	elif x <= 1e+195:
		tmp = 0.5 * y
	else:
		tmp = x / (2.0 * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.5e+18)
		tmp = Float64(0.5 * y);
	elseif (x <= 5.5e+182)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	elseif (x <= 1e+195)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x / Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.5e+18)
		tmp = 0.5 * y;
	elseif (x <= 5.5e+182)
		tmp = (x / y) * (x / 2.0);
	elseif (x <= 1e+195)
		tmp = 0.5 * y;
	else
		tmp = x / (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.5e+18], N[(0.5 * y), $MachinePrecision], If[LessEqual[x, 5.5e+182], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+195], N[(0.5 * y), $MachinePrecision], N[(x / N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.5e18 or 5.49999999999999977e182 < x < 9.99999999999999977e194

   1. Initial program 68.8%

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

   2. Taylor expanded in y around inf 41.3%

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

   if 3.5e18 < x < 5.49999999999999977e182

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf 48.4%

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

      1. unpow2 48.4%

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

      2. times-frac 51.3%

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

   4. Applied egg-rr 51.3%

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

   if 9.99999999999999977e194 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 84.6%

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

      4. metadata-eval 84.6%

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

      5. div-inv 84.6%

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

      6. clear-num 84.6%

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

   4. Applied egg-rr 84.6%

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

      1. div-inv 84.6%

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

      2. associate-*r* 84.6%

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

      3. *-commutative 84.6%

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

      4. associate-/r/ 84.7%

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

      5. div-inv 84.8%

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

      6. *-un-lft-identity 84.8%

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

      7. *-commutative 84.8%

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

      8. times-frac 84.8%

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

      9. metadata-eval 84.8%

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

   6. Applied egg-rr 84.8%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 11: 34.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

2. Taylor expanded in y around inf 35.7%

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

3. Final simplification 35.7%

   \[\leadsto 0.5 \cdot y \]

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 2023301 
(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)))