2-ancestry mixing, zero discriminant

Percentage Accurate: 76.6% → 98.7%

Time: 4.6s

Alternatives: 7

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{a + a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt g) (cbrt (+ a a))))

C

double code(double g, double a) {
	return cbrt(g) / cbrt((a + a));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) / Math.cbrt((a + a));
}

Julia

function code(g, a)
	return Float64(cbrt(g) / cbrt(Float64(a + a)))
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[N[(a + a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

   3. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   5. lower-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}} \]

   6. lower-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a}}} \]

   8. count-2-rev N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

   9. lower-+.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

3. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a + a}}} \]
4. Add Preprocessing

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-104}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, -0.3333333333333333 \cdot \log a\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log g \cdot 0.3333333333333333 - \log \left(a + a\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a)))))
   (if (<= t_0 4e-104)
     (exp
      (fma (log (* 0.5 g)) 0.3333333333333333 (* -0.3333333333333333 (log a))))
     (if (<= t_0 1e+103)
       (cbrt (/ 1.0 (/ (+ a a) g)))
       (exp
        (-
         (* (log g) 0.3333333333333333)
         (* (log (+ a a)) 0.3333333333333333)))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = exp(fma(log((0.5 * g)), 0.3333333333333333, (-0.3333333333333333 * log(a))));
	} else if (t_0 <= 1e+103) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = exp(((log(g) * 0.3333333333333333) - (log((a + a)) * 0.3333333333333333)));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	tmp = 0.0
	if (t_0 <= 4e-104)
		tmp = exp(fma(log(Float64(0.5 * g)), 0.3333333333333333, Float64(-0.3333333333333333 * log(a))));
	elseif (t_0 <= 1e+103)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = exp(Float64(Float64(log(g) * 0.3333333333333333) - Float64(log(Float64(a + a)) * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-104], N[Exp[N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(-0.3333333333333333 * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+103], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[Exp[N[(N[(N[Log[g], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.99999999999999971e-104

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{\frac{1}{3}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      6. lower-unsound-log.f64 35.9

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      8. count-2-rev N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      9. lower-+.f64 35.9

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      3. mult-flip N/A

         \[\leadsto e^{\log \color{blue}{\left(g \cdot \frac{1}{a + a}\right)} \cdot \frac{1}{3}} \]

      4. lift-+.f64 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      5. count-2 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      6. associate-/r* N/A

         \[\leadsto e^{\log \left(g \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right) \cdot \frac{1}{3}} \]

      7. metadata-eval N/A

         \[\leadsto e^{\log \left(g \cdot \frac{\color{blue}{\frac{1}{2}}}{a}\right) \cdot \frac{1}{3}} \]

      8. associate-/l* N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g \cdot \frac{1}{2}}{a}\right)} \cdot \frac{1}{3}} \]

      9. *-commutative N/A

         \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      11. log-div N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      12. lower-unsound--.f64 N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto e^{\left(\color{blue}{\log \left(\frac{1}{2} \cdot g\right)} - \log a\right) \cdot \frac{1}{3}} \]

      14. lower-unsound-log.f64 22.6

          \[\leadsto e^{\left(\log \left(0.5 \cdot g\right) - \color{blue}{\log a}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\left(\log \left(0.5 \cdot g\right) - \log a\right)} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right) \cdot \frac{1}{3}}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\frac{1}{3} \cdot \left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)}} \]

      4. sub-flip N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) + \left(\mathsf{neg}\left(\log a\right)\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot g\right) \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\log a\right)\right) \cdot \frac{1}{3}}} \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\frac{1}{2} \cdot g\right) \cdot \frac{1}{3} + \color{blue}{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\log a\right)\right)}} \]

      7. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \left(\mathsf{neg}\left(\log a\right)\right)\right)}} \]

      8. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \left(\mathsf{neg}\left(\color{blue}{\log a}\right)\right)\right)} \]

      9. neg-log N/A

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \color{blue}{\log \left(\frac{1}{a}\right)}\right)} \]

      10. log-pow-rev N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\log \left({\left(\frac{1}{a}\right)}^{\frac{1}{3}}\right)}\right)} \]

      11. pow1/3 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \color{blue}{\left(\sqrt[3]{\frac{1}{a}}\right)}\right)} \]

      12. inv-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left(\sqrt[3]{\color{blue}{{a}^{-1}}}\right)\right)} \]

      13. cbrt-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \color{blue}{\left({a}^{\left(\frac{-1}{3}\right)}\right)}\right)} \]

      14. metadata-eval N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left({a}^{\color{blue}{\frac{-1}{3}}}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left({a}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}\right)\right)} \]

      16. log-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \log a}\right)} \]

      17. lower-unsound-log.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \color{blue}{\log a}\right)} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \log a}\right)} \]

      19. metadata-eval 22.6

          \[\leadsto e^{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, \color{blue}{-0.3333333333333333} \cdot \log a\right)} \]

   7. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, -0.3333333333333333 \cdot \log a\right)}} \]

   if 3.99999999999999971e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

      2. div-flip N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      4. lower-unsound-/.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{2 \cdot a}{g}}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{g}}} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

      7. lower-+.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

   3. Applied rewrites 76.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]

   if 1e103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{\frac{1}{3}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      6. lower-unsound-log.f64 35.9

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      8. count-2-rev N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      9. lower-+.f64 35.9

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      3. log-div N/A

         \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot \frac{1}{3}} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot \frac{1}{3}} \]

      5. lower-unsound-log.f64 N/A

         \[\leadsto e^{\left(\color{blue}{\log g} - \log \left(a + a\right)\right) \cdot \frac{1}{3}} \]

      6. lower-unsound-log.f64 22.6

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right) \cdot \frac{1}{3}}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\frac{1}{3} \cdot \left(\log g - \log \left(a + a\right)\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log g - \log \left(a + a\right)\right)}} \]

      4. sub-flip N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log g + \left(\mathsf{neg}\left(\log \left(a + a\right)\right)\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{\color{blue}{\log g \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\log \left(a + a\right)\right)\right) \cdot \frac{1}{3}}} \]

      6. fp-cancel-sub-sign N/A

         \[\leadsto e^{\color{blue}{\log g \cdot \frac{1}{3} - \log \left(a + a\right) \cdot \frac{1}{3}}} \]

      7. lower--.f64 N/A

         \[\leadsto e^{\color{blue}{\log g \cdot \frac{1}{3} - \log \left(a + a\right) \cdot \frac{1}{3}}} \]

      8. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log g \cdot \frac{1}{3}} - \log \left(a + a\right) \cdot \frac{1}{3}} \]

      9. lower-*.f64 22.6

         \[\leadsto e^{\log g \cdot 0.3333333333333333 - \color{blue}{\log \left(a + a\right) \cdot 0.3333333333333333}} \]

   7. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\log g \cdot 0.3333333333333333 - \log \left(a + a\right) \cdot 0.3333333333333333}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, -0.3333333333333333 \cdot \log a\right)}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1
         (exp
          (fma
           (log (* 0.5 g))
           0.3333333333333333
           (* -0.3333333333333333 (log a))))))
   (if (<= t_0 4e-104)
     t_1
     (if (<= t_0 1e+103) (cbrt (/ 1.0 (/ (+ a a) g))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(fma(log((0.5 * g)), 0.3333333333333333, (-0.3333333333333333 * log(a))));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(fma(log(Float64(0.5 * g)), 0.3333333333333333, Float64(-0.3333333333333333 * log(a))))
	tmp = 0.0
	if (t_0 <= 4e-104)
		tmp = t_1;
	elseif (t_0 <= 1e+103)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(-0.3333333333333333 * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-104], t$95$1, If[LessEqual[t$95$0, 1e+103], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.99999999999999971e-104 or 1e103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{\frac{1}{3}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      6. lower-unsound-log.f64 35.9

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      8. count-2-rev N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      9. lower-+.f64 35.9

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      3. mult-flip N/A

         \[\leadsto e^{\log \color{blue}{\left(g \cdot \frac{1}{a + a}\right)} \cdot \frac{1}{3}} \]

      4. lift-+.f64 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      5. count-2 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      6. associate-/r* N/A

         \[\leadsto e^{\log \left(g \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right) \cdot \frac{1}{3}} \]

      7. metadata-eval N/A

         \[\leadsto e^{\log \left(g \cdot \frac{\color{blue}{\frac{1}{2}}}{a}\right) \cdot \frac{1}{3}} \]

      8. associate-/l* N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g \cdot \frac{1}{2}}{a}\right)} \cdot \frac{1}{3}} \]

      9. *-commutative N/A

         \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      11. log-div N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      12. lower-unsound--.f64 N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto e^{\left(\color{blue}{\log \left(\frac{1}{2} \cdot g\right)} - \log a\right) \cdot \frac{1}{3}} \]

      14. lower-unsound-log.f64 22.6

          \[\leadsto e^{\left(\log \left(0.5 \cdot g\right) - \color{blue}{\log a}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\left(\log \left(0.5 \cdot g\right) - \log a\right)} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right) \cdot \frac{1}{3}}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\frac{1}{3} \cdot \left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)}} \]

      4. sub-flip N/A

         \[\leadsto e^{\frac{1}{3} \cdot \color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) + \left(\mathsf{neg}\left(\log a\right)\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot g\right) \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\log a\right)\right) \cdot \frac{1}{3}}} \]

      6. *-commutative N/A

         \[\leadsto e^{\log \left(\frac{1}{2} \cdot g\right) \cdot \frac{1}{3} + \color{blue}{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\log a\right)\right)}} \]

      7. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \left(\mathsf{neg}\left(\log a\right)\right)\right)}} \]

      8. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \left(\mathsf{neg}\left(\color{blue}{\log a}\right)\right)\right)} \]

      9. neg-log N/A

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \frac{1}{3} \cdot \color{blue}{\log \left(\frac{1}{a}\right)}\right)} \]

      10. log-pow-rev N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\log \left({\left(\frac{1}{a}\right)}^{\frac{1}{3}}\right)}\right)} \]

      11. pow1/3 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \color{blue}{\left(\sqrt[3]{\frac{1}{a}}\right)}\right)} \]

      12. inv-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left(\sqrt[3]{\color{blue}{{a}^{-1}}}\right)\right)} \]

      13. cbrt-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \color{blue}{\left({a}^{\left(\frac{-1}{3}\right)}\right)}\right)} \]

      14. metadata-eval N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left({a}^{\color{blue}{\frac{-1}{3}}}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \log \left({a}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}\right)\right)} \]

      16. log-pow N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \log a}\right)} \]

      17. lower-unsound-log.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \color{blue}{\log a}\right)} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(\log \left(\frac{1}{2} \cdot g\right), \frac{1}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \log a}\right)} \]

      19. metadata-eval 22.6

          \[\leadsto e^{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, \color{blue}{-0.3333333333333333} \cdot \log a\right)} \]

   7. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(0.5 \cdot g\right), 0.3333333333333333, -0.3333333333333333 \cdot \log a\right)}} \]

   if 3.99999999999999971e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

      2. div-flip N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      4. lower-unsound-/.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{2 \cdot a}{g}}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{g}}} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

      7. lower-+.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

   3. Applied rewrites 76.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 42.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log \left(0.5 \cdot g\right) - \log a\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1 (exp (* (- (log (* 0.5 g)) (log a)) 0.3333333333333333))))
   (if (<= t_0 4e-104)
     t_1
     (if (<= t_0 1e+103) (cbrt (/ 1.0 (/ (+ a a) g))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(((log((0.5 * g)) - log(a)) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.exp(((Math.log((0.5 * g)) - Math.log(a)) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = Math.cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(Float64(Float64(log(Float64(0.5 * g)) - log(a)) * 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= 4e-104)
		tmp = t_1;
	elseif (t_0 <= 1e+103)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[N[(0.5 * g), $MachinePrecision]], $MachinePrecision] - N[Log[a], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-104], t$95$1, If[LessEqual[t$95$0, 1e+103], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.99999999999999971e-104 or 1e103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{\frac{1}{3}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      6. lower-unsound-log.f64 35.9

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      8. count-2-rev N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      9. lower-+.f64 35.9

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      3. mult-flip N/A

         \[\leadsto e^{\log \color{blue}{\left(g \cdot \frac{1}{a + a}\right)} \cdot \frac{1}{3}} \]

      4. lift-+.f64 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      5. count-2 N/A

         \[\leadsto e^{\log \left(g \cdot \frac{1}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      6. associate-/r* N/A

         \[\leadsto e^{\log \left(g \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right) \cdot \frac{1}{3}} \]

      7. metadata-eval N/A

         \[\leadsto e^{\log \left(g \cdot \frac{\color{blue}{\frac{1}{2}}}{a}\right) \cdot \frac{1}{3}} \]

      8. associate-/l* N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g \cdot \frac{1}{2}}{a}\right)} \cdot \frac{1}{3}} \]

      9. *-commutative N/A

         \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{2} \cdot g}}{a}\right) \cdot \frac{1}{3}} \]

      11. log-div N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      12. lower-unsound--.f64 N/A

          \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2} \cdot g\right) - \log a\right)} \cdot \frac{1}{3}} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto e^{\left(\color{blue}{\log \left(\frac{1}{2} \cdot g\right)} - \log a\right) \cdot \frac{1}{3}} \]

      14. lower-unsound-log.f64 22.6

          \[\leadsto e^{\left(\log \left(0.5 \cdot g\right) - \color{blue}{\log a}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\left(\log \left(0.5 \cdot g\right) - \log a\right)} \cdot 0.3333333333333333} \]

   if 3.99999999999999971e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

      2. div-flip N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      4. lower-unsound-/.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{2 \cdot a}{g}}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{g}}} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

      7. lower-+.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

   3. Applied rewrites 76.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 42.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{g}{2 \cdot a}}\\ t_1 := e^{\left(\log g - \log \left(a + a\right)\right) \cdot 0.3333333333333333}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{a + a}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (/ g (* 2.0 a))))
        (t_1 (exp (* (- (log g) (log (+ a a))) 0.3333333333333333))))
   (if (<= t_0 4e-104)
     t_1
     (if (<= t_0 1e+103) (cbrt (/ 1.0 (/ (+ a a) g))) t_1))))

C

double code(double g, double a) {
	double t_0 = cbrt((g / (2.0 * a)));
	double t_1 = exp(((log(g) - log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g / (2.0 * a)));
	double t_1 = Math.exp(((Math.log(g) - Math.log((a + a))) * 0.3333333333333333));
	double tmp;
	if (t_0 <= 4e-104) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = Math.cbrt((1.0 / ((a + a) / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g / Float64(2.0 * a)))
	t_1 = exp(Float64(Float64(log(g) - log(Float64(a + a))) * 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= 4e-104)
		tmp = t_1;
	elseif (t_0 <= 1e+103)
		tmp = cbrt(Float64(1.0 / Float64(Float64(a + a) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[Log[g], $MachinePrecision] - N[Log[N[(a + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-104], t$95$1, If[LessEqual[t$95$0, 1e+103], N[Power[N[(1.0 / N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 3.99999999999999971e-104 or 1e103 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a)))

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{\frac{1}{3}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right) \cdot \frac{1}{3}}} \]

      6. lower-unsound-log.f64 35.9

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{2 \cdot a}\right)} \cdot 0.3333333333333333} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

      8. count-2-rev N/A

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

      9. lower-+.f64 35.9

         \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot 0.3333333333333333} \]

   3. Applied rewrites 35.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot 0.3333333333333333}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

      3. log-div N/A

         \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot \frac{1}{3}} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot \frac{1}{3}} \]

      5. lower-unsound-log.f64 N/A

         \[\leadsto e^{\left(\color{blue}{\log g} - \log \left(a + a\right)\right) \cdot \frac{1}{3}} \]

      6. lower-unsound-log.f64 22.6

         \[\leadsto e^{\left(\log g - \color{blue}{\log \left(a + a\right)}\right) \cdot 0.3333333333333333} \]

   5. Applied rewrites 22.6%

      \[\leadsto e^{\color{blue}{\left(\log g - \log \left(a + a\right)\right)} \cdot 0.3333333333333333} \]

   if 3.99999999999999971e-104 < (cbrt.f64 (/.f64 g (*.f64 #s(literal 2 binary64) a))) < 1e103

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

      2. div-flip N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

      4. lower-unsound-/.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{2 \cdot a}{g}}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{g}}} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

      7. lower-+.f64 76.0

         \[\leadsto \sqrt[3]{\frac{1}{\frac{\color{blue}{a + a}}{g}}} \]

   3. Applied rewrites 76.0%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{a + a}{g}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a + a}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (cbrt (/ (+ a a) g))))

C

double code(double g, double a) {
	return 1.0 / cbrt(((a + a) / g));
}

Java

public static double code(double g, double a) {
	return 1.0 / Math.cbrt(((a + a) / g));
}

Julia

function code(g, a)
	return Float64(1.0 / cbrt(Float64(Float64(a + a) / g)))
end

Wolfram

code[g_, a_] := N[(1.0 / N[Power[N[(N[(a + a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

   3. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   5. lower-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}} \]

   6. lower-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a}}} \]

   8. count-2-rev N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

   9. lower-+.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a + a}}} \]

3. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a + a}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a + a}}} \]

   2. lift-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{a + a}} \]

   3. lift-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{a + a}}} \]

   4. cbrt-undiv N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a + a}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a + a}}} \]

   6. pow1/3 N/A

      \[\leadsto \color{blue}{{\left(\frac{g}{a + a}\right)}^{\frac{1}{3}}} \]

   7. pow-to-exp N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{g}{a + a}\right) \cdot \frac{1}{3}}} \]

   8. lift-/.f64 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a + a}\right)} \cdot \frac{1}{3}} \]

   9. lift-+.f64 N/A

      \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a + a}}\right) \cdot \frac{1}{3}} \]

   10. count-2 N/A

       \[\leadsto e^{\log \left(\frac{g}{\color{blue}{2 \cdot a}}\right) \cdot \frac{1}{3}} \]

   11. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{g}{\color{blue}{a \cdot 2}}\right) \cdot \frac{1}{3}} \]

   12. associate-/r* N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{g}{a}}{2}\right)} \cdot \frac{1}{3}} \]

   13. lift-/.f64 N/A

       \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{g}{a}}}{2}\right) \cdot \frac{1}{3}} \]

   14. mult-flip-rev N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{g}{a} \cdot \frac{1}{2}\right)} \cdot \frac{1}{3}} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{g}{a} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{1}{3}} \]

   16. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{g}{a} \cdot \color{blue}{\frac{2}{4}}\right) \cdot \frac{1}{3}} \]

   17. associate-/l* N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{g}{a} \cdot 2}{4}\right)} \cdot \frac{1}{3}} \]

   18. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\color{blue}{2 \cdot \frac{g}{a}}}{4}\right) \cdot \frac{1}{3}} \]

   19. lift-*.f64 N/A

       \[\leadsto e^{\log \left(\frac{\color{blue}{2 \cdot \frac{g}{a}}}{4}\right) \cdot \frac{1}{3}} \]

   20. pow-to-exp N/A

       \[\leadsto \color{blue}{{\left(\frac{2 \cdot \frac{g}{a}}{4}\right)}^{\frac{1}{3}}} \]

   21. pow1/3 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\frac{2 \cdot \frac{g}{a}}{4}}} \]

   22. cbrt-undiv N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{2 \cdot \frac{g}{a}}}{\sqrt[3]{4}}} \]

   23. lift-cbrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{2 \cdot \frac{g}{a}}}}{\sqrt[3]{4}} \]

   24. lift-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{2 \cdot \frac{g}{a}}}{\color{blue}{\sqrt[3]{4}}} \]

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g + g} \cdot \sqrt[3]{0.25}}{\sqrt[3]{a}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g + g} \cdot \sqrt[3]{\frac{1}{4}}}{\sqrt[3]{a}}} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{g + g} \cdot \sqrt[3]{\frac{1}{4}}\right) \cdot 1}}{\sqrt[3]{a}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{g + g} \cdot \sqrt[3]{\frac{1}{4}}\right)} \cdot 1}{\sqrt[3]{a}} \]

   4. lift-cbrt.f64 N/A

      \[\leadsto \frac{\left(\color{blue}{\sqrt[3]{g + g}} \cdot \sqrt[3]{\frac{1}{4}}\right) \cdot 1}{\sqrt[3]{a}} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\left(\sqrt[3]{g + g} \cdot \color{blue}{\sqrt[3]{\frac{1}{4}}}\right) \cdot 1}{\sqrt[3]{a}} \]

   6. cbrt-unprod N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(g + g\right) \cdot \frac{1}{4}}} \cdot 1}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{\left(g + g\right) \cdot \frac{1}{4}} \cdot \color{blue}{\sqrt[3]{1}}}{\sqrt[3]{a}} \]

   8. cbrt-unprod N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(g + g\right) \cdot \frac{1}{4}\right) \cdot 1}}}{\sqrt[3]{a}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\left(g + g\right)} \cdot \frac{1}{4}\right) \cdot 1}}{\sqrt[3]{a}} \]

   10. count-2 N/A

       \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\left(2 \cdot g\right)} \cdot \frac{1}{4}\right) \cdot 1}}{\sqrt[3]{a}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\left(g \cdot 2\right)} \cdot \frac{1}{4}\right) \cdot 1}}{\sqrt[3]{a}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \left(2 \cdot \frac{1}{4}\right)\right)} \cdot 1}}{\sqrt[3]{a}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{1}{2}}\right) \cdot 1}}{\sqrt[3]{a}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{1}{2}}\right) \cdot 1}}{\sqrt[3]{a}} \]

   15. mult-flip-rev N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{2}} \cdot 1}}{\sqrt[3]{a}} \]

   16. *-rgt-identity N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   17. lift-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\frac{g}{2}}}{\color{blue}{\sqrt[3]{a}}} \]

   18. cbrt-div N/A

       \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{g}{2}}{a}}} \]

   19. associate-/r* N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

   20. count-2 N/A

       \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   21. lift-+.f64 N/A

       \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   22. lift-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a + a}}} \]

   23. rem-exp-log N/A

       \[\leadsto \sqrt[3]{\color{blue}{e^{\log \left(\frac{g}{a + a}\right)}}} \]

7. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a + a}{g}}}} \]
8. Add Preprocessing

Alternative 7: 42.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a + a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (+ a a))))

C

double code(double g, double a) {
	return cbrt((g / (a + a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (a + a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(a + a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{2 \cdot a}}} \]

   2. count-2-rev N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   3. lower-+.f64 76.6

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

3. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a + a}}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025164 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))