2-ancestry mixing, positive discriminant

Percentage Accurate: 44.7% → 97.2%

Time: 7.7s

Alternatives: 5

Speedup: 3.9×

Specification

Math

\[\begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 44.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 97.2% accurate, 0.5× speedup

Math

\[\mathsf{fma}\left(-1, \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}, -1 \cdot \frac{{\left(\left|h\right|\right)}^{0.6666666666666666} \cdot {\left(\sqrt[3]{0.5}\right)}^{2}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right) \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  -1.0
  (/ (* (cbrt g) (* (cbrt 0.5) (cbrt 2.0))) (cbrt a))
  (*
   -1.0
   (/
    (* (pow (fabs h) 0.6666666666666666) (pow (cbrt 0.5) 2.0))
    (* (cbrt a) (cbrt g))))))

C

double code(double g, double h, double a) {
	return fma(-1.0, ((cbrt(g) * (cbrt(0.5) * cbrt(2.0))) / cbrt(a)), (-1.0 * ((pow(fabs(h), 0.6666666666666666) * pow(cbrt(0.5), 2.0)) / (cbrt(a) * cbrt(g)))));
}

Julia

function code(g, h, a)
	return fma(-1.0, Float64(Float64(cbrt(g) * Float64(cbrt(0.5) * cbrt(2.0))) / cbrt(a)), Float64(-1.0 * Float64(Float64((abs(h) ^ 0.6666666666666666) * (cbrt(0.5) ^ 2.0)) / Float64(cbrt(a) * cbrt(g)))))
end

Wolfram

code[g_, h_, a_] := N[(-1.0 * N[(N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(N[Power[N[Abs[h], $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[Power[0.5, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[a, 1/3], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

2. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 28.1

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

4. Applied rewrites 28.1%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
5. Step-by-step derivation

6. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a + a}} + \sqrt[3]{\frac{-g}{a}}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + \left(\mathsf{neg}\left(g\right)\right)}}{a + a}} + \sqrt[3]{\frac{-g}{a}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)}} + \left(\mathsf{neg}\left(g\right)\right)}{a + a}} + \sqrt[3]{\frac{-g}{a}} \]

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. difference-of-squares N/A

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

   10. pow1/2 N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\sqrt{g \cdot g - h \cdot h}} + \left(\mathsf{neg}\left(g\right)\right)}{a + a}} + \sqrt[3]{\frac{-g}{a}} \]

   11. +-commutative N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}}}{a + a}} + \sqrt[3]{\frac{-g}{a}} \]

   12. sum-to-mult N/A

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

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

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} + -1 \cdot \frac{{h}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{\frac{1}{2}}\right)}^{2}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}} \]
10. Step-by-step derivation

    1. lower-fma.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cbrt.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-cbrt.f64 N/A

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

    7. lower-cbrt.f64 N/A

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

    8. lower-cbrt.f64 N/A

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

    9. lower-*.f64 N/A

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

11. Applied rewrites 47.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}, -1 \cdot \frac{{h}^{0.6666666666666666} \cdot {\left(\sqrt[3]{0.5}\right)}^{2}}{\sqrt[3]{a} \cdot \sqrt[3]{g}}\right)} \]
12. Add Preprocessing

Alternative 2: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|a\right|\right)}^{\left(\frac{-1}{2}\right)}\\ \mathsf{copysign}\left(1, a\right) \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{-t\_0 \cdot t\_0}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (pow (fabs a) (/ -1.0 2.0))))
   (* (copysign 1.0 a) (* (cbrt g) (cbrt (- (* t_0 t_0)))))))

C

double code(double g, double h, double a) {
	double t_0 = pow(fabs(a), (-1.0 / 2.0));
	return copysign(1.0, a) * (cbrt(g) * cbrt(-(t_0 * t_0)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.pow(Math.abs(a), (-1.0 / 2.0));
	return Math.copySign(1.0, a) * (Math.cbrt(g) * Math.cbrt(-(t_0 * t_0)));
}

Julia

function code(g, h, a)
	t_0 = abs(a) ^ Float64(-1.0 / 2.0)
	return Float64(copysign(1.0, a) * Float64(cbrt(g) * cbrt(Float64(-Float64(t_0 * t_0)))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[Abs[a], $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[(-N[(t$95$0 * t$95$0), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

2. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]

4. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   7. cbrt-neg N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   9. metadata-eval 95.9

      \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   10. lower-*.f64 N/A

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

   11. lift-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   14. cbrt-neg N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   16. cbrt-unprod N/A

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

   17. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{-1 \cdot g}}{\sqrt[3]{a}} \]

   18. mul-1-neg N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}} \]

   19. cbrt-neg-rev N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   20. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}} \]

   21. lower-neg.f64 95.9

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

6. Applied rewrites 95.9%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   3. lift-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}} \]

   4. cbrt-neg-rev N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a}}} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}} \]

   6. cbrt-undiv N/A

      \[\leadsto \sqrt[3]{\frac{\mathsf{neg}\left(g\right)}{a}} \]

   7. distribute-frac-neg N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   10. mult-flip N/A

       \[\leadsto \sqrt[3]{\mathsf{neg}\left(g \cdot \frac{1}{a}\right)} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt[3]{g \cdot \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

   12. cbrt-prod N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)} \]

   16. lower-neg.f64 N/A

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-\frac{1}{a}} \]

   17. lower-/.f64 95.9

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-\frac{1}{a}} \]

8. Applied rewrites 95.9%

   \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\sqrt[3]{-\frac{1}{a}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-\frac{1}{a}} \]

   2. inv-pow N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-{a}^{-1}} \]

   3. sqr-pow N/A

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

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

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

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

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

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

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

   7. lower-unsound-pow.f64 N/A

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

   8. lower-unsound-/.f64 48.2

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

10. Applied rewrites 48.2%

    \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-{a}^{\left(\frac{-1}{2}\right)} \cdot {a}^{\left(\frac{-1}{2}\right)}} \]
11. Add Preprocessing

Alternative 3: 95.9% accurate, 2.1× speedup

Math

\[\sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{a}} \]

FPCore

(FPCore (g h a) :precision binary64 (* (cbrt g) (cbrt (/ -1.0 a))))

C

double code(double g, double h, double a) {
	return cbrt(g) * cbrt((-1.0 / a));
}

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(g) * cbrt(Float64(-1.0 / a)))
end

Wolfram

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

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

2. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]

4. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   7. cbrt-neg N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   9. metadata-eval 95.9

      \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   10. lower-*.f64 N/A

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

   11. lift-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   14. cbrt-neg N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   16. cbrt-unprod N/A

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

   17. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{-1 \cdot g}}{\sqrt[3]{a}} \]

   18. mul-1-neg N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}} \]

   19. cbrt-neg-rev N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   20. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}} \]

   21. lower-neg.f64 95.9

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

6. Applied rewrites 95.9%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   3. lift-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}} \]

   4. cbrt-neg-rev N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a}}} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}} \]

   6. cbrt-undiv N/A

      \[\leadsto \sqrt[3]{\frac{\mathsf{neg}\left(g\right)}{a}} \]

   7. distribute-frac-neg N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\mathsf{neg}\left(\frac{g}{a}\right)} \]

   10. mult-flip N/A

       \[\leadsto \sqrt[3]{\mathsf{neg}\left(g \cdot \frac{1}{a}\right)} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt[3]{g \cdot \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

   12. cbrt-prod N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)} \]

   16. lower-neg.f64 N/A

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-\frac{1}{a}} \]

   17. lower-/.f64 95.9

       \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{-\frac{1}{a}} \]

8. Applied rewrites 95.9%

   \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\sqrt[3]{-\frac{1}{a}}} \]
9. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{\mathsf{neg}\left(1\right)}{a}} \]

   4. metadata-eval N/A

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{a}} \]

   5. lower-/.f64 95.9

      \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{a}} \]

10. Applied rewrites 95.9%

    \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{a}} \]
11. Add Preprocessing

Alternative 4: 95.9% accurate, 2.3× speedup

Math

\[\frac{\sqrt[3]{g}}{-\sqrt[3]{a}} \]

FPCore

(FPCore (g h a) :precision binary64 (/ (cbrt g) (- (cbrt a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

2. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]

4. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   7. cbrt-neg N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   9. metadata-eval 95.9

      \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   10. lower-*.f64 N/A

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

   11. lift-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot -1}{\sqrt[3]{a}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\sqrt[3]{a}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\sqrt[3]{a}} \]

   14. cbrt-neg N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\sqrt[3]{a}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\sqrt[3]{a}} \]

   16. cbrt-unprod N/A

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

   17. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{-1 \cdot g}}{\sqrt[3]{a}} \]

   18. mul-1-neg N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a}} \]

   19. cbrt-neg-rev N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   20. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}} \]

   21. lower-neg.f64 95.9

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

6. Applied rewrites 95.9%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{\color{blue}{a}}} \]

   3. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}}\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{a}\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{a}\right)}} \]

   6. lower-neg.f64 95.9

      \[\leadsto \frac{\sqrt[3]{g}}{-\sqrt[3]{a}} \]

8. Applied rewrites 95.9%

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

Alternative 5: 73.7% accurate, 3.9× speedup

Math

\[-\sqrt[3]{\frac{g}{a}} \]

FPCore

(FPCore (g h a) :precision binary64 (- (cbrt (/ g a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.7%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

2. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.2

      \[\leadsto \frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}} \]

4. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   6. lift-cbrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{2}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   7. cbrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{2} \cdot 2}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   10. cbrt-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot -1}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot -1}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   14. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot -1}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \left(\mathsf{neg}\left(\sqrt[3]{1}\right)\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   17. cbrt-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \sqrt[3]{\mathsf{neg}\left(1\right)}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g} \cdot \sqrt[3]{-1}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   19. cbrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{g \cdot -1}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{-1 \cdot g}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   21. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   22. cbrt-neg-rev N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

   23. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\mathsf{neg}\left(\sqrt[3]{a}\right)}\right) \]

6. Applied rewrites 73.7%

   \[\leadsto -\sqrt[3]{\frac{g}{a}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025181 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))