2-ancestry mixing, positive discriminant

Percentage Accurate: 43.3% → 95.9%

Time: 10.1s

Alternatives: 5

Speedup: 1.9×

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: 43.3% 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: 95.9% accurate, 1.0× speedup

Math

\[\frac{\sqrt[3]{\frac{\left(g + 0\right) - g}{a + a} \cdot \left(a \cdot -2\right)} + \sqrt[3]{\left(g + 0\right) + g}}{\sqrt[3]{a \cdot -2}} \]

FPCore

(FPCore (g h a)
  :precision binary64
  (/
 (+
  (cbrt (* (/ (- (+ g 0.0) g) (+ a a)) (* a -2.0)))
  (cbrt (+ (+ g 0.0) g)))
 (cbrt (* a -2.0))))

C

double code(double g, double h, double a) {
	return (cbrt(((((g + 0.0) - g) / (a + a)) * (a * -2.0))) + cbrt(((g + 0.0) + g))) / cbrt((a * -2.0));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(((((g + 0.0) - g) / (a + a)) * (a * -2.0))) + Math.cbrt(((g + 0.0) + g))) / Math.cbrt((a * -2.0));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(Float64(Float64(Float64(g + 0.0) - g) / Float64(a + a)) * Float64(a * -2.0))) + cbrt(Float64(Float64(g + 0.0) + g))) / cbrt(Float64(a * -2.0)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(N[(N[(N[(g + 0.0), $MachinePrecision] - g), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision] * N[(a * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + 0.0), $MachinePrecision] + g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.3%

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

3. Applied rewrites 46.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right)} \]

4. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 32.2%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

6. Applied rewrites 32.2%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

7. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 95.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

9. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
10. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. associate-/l/ N/A

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

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

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

    8. count-2-rev N/A

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

    9. lift-+.f64 N/A

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

    10. cbrt-div N/A

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

    11. metadata-eval N/A

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

    12. cbrt-neg-rev N/A

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

    13. lower-/.f64 N/A

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

    14. cbrt-neg-rev N/A

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

    15. lower-cbrt.f64 N/A

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

    16. lift-+.f64 N/A

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

    17. count-2-rev N/A

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

    18. distribute-lft-neg-in N/A

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

    19. lower-*.f64 N/A

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

    20. metadata-eval 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{\color{blue}{-2} \cdot a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

11. Applied rewrites 95.7%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{-2 \cdot a}}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

13. Applied rewrites 95.7%

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\left(g + 0\right) - g}{a + a} \cdot \left(a \cdot -2\right)} + \sqrt[3]{\left(g + 0\right) + g}}{\sqrt[3]{a \cdot -2}}} \]
14. Add Preprocessing

Alternative 2: 95.7% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

FPCore

(FPCore (g h a)
  :precision binary64
  (fma
 (/ 1.0 (cbrt (* -2.0 a)))
 (cbrt (+ (+ g 0.0) g))
 (cbrt (/ (- (+ g 0.0) g) (+ a a)))))

C

double code(double g, double h, double a) {
	return fma((1.0 / cbrt((-2.0 * a))), cbrt(((g + 0.0) + g)), cbrt((((g + 0.0) - g) / (a + a))));
}

Julia

function code(g, h, a)
	return fma(Float64(1.0 / cbrt(Float64(-2.0 * a))), cbrt(Float64(Float64(g + 0.0) + g)), cbrt(Float64(Float64(Float64(g + 0.0) - g) / Float64(a + a))))
end

Wolfram

code[g_, h_, a_] := N[(N[(1.0 / N[Power[N[(-2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(g + 0.0), $MachinePrecision] + g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(N[(g + 0.0), $MachinePrecision] - g), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.3%

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

3. Applied rewrites 46.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right)} \]

4. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 32.2%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

6. Applied rewrites 32.2%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

7. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 95.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

9. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
10. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. associate-/l/ N/A

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

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

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

    8. count-2-rev N/A

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

    9. lift-+.f64 N/A

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

    10. cbrt-div N/A

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

    11. metadata-eval N/A

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

    12. cbrt-neg-rev N/A

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

    13. lower-/.f64 N/A

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

    14. cbrt-neg-rev N/A

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

    15. lower-cbrt.f64 N/A

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

    16. lift-+.f64 N/A

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

    17. count-2-rev N/A

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

    18. distribute-lft-neg-in N/A

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

    19. lower-*.f64 N/A

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

    20. metadata-eval 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{\color{blue}{-2} \cdot a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

11. Applied rewrites 95.7%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{-2 \cdot a}}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. associate-*r/ N/A

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

    7. sum-to-mult-rev N/A

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

    8. *-commutative N/A

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

    9. lift-+.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. distribute-rgt1-in N/A

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

    12. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + \left(0 \cdot h\right) \cdot \frac{1}{2}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    13. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0 \cdot \frac{1}{2}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    15. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0 \cdot \color{blue}{h}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    16. metadata-eval N/A

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

    17. distribute-rgt1-in N/A

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

    18. lift-*.f64 N/A

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

    19. lift-+.f64 N/A

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

    20. lower-+.f64 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + \color{blue}{\left(h + -1 \cdot h\right)}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    21. lift-+.f64 N/A

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

    22. lift-*.f64 N/A

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

    23. distribute-rgt1-in N/A

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

    24. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0 \cdot h\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    25. mul0-lft 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

13. Applied rewrites 95.7%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + \color{blue}{0}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]
14. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot \color{blue}{g} - g}{a + a}}\right) \]

    3. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    5. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{\frac{1}{2} \cdot \left(h + -1 \cdot h\right)}{g}\right) \cdot g - g}{a + a}}\right) \]

    7. sum-to-mult-rev N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{\frac{1}{2} \cdot \left(h + -1 \cdot h\right)}\right) - g}{a + a}}\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \color{blue}{\frac{1}{2}}\right) - g}{a + a}}\right) \]

    9. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    11. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(\left(-1 + 1\right) \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(0 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    13. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

    15. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot \color{blue}{h}\right) - g}{a + a}}\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(-1 + 1\right) \cdot h\right) - g}{a + a}}\right) \]

    17. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    18. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot \color{blue}{h}\right)\right) - g}{a + a}}\right) \]

    19. lift-+.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    20. lower-+.f64 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{\left(h + -1 \cdot h\right)}\right) - g}{a + a}}\right) \]

    21. lift-+.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot \color{blue}{h}\right)\right) - g}{a + a}}\right) \]

    23. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(-1 + 1\right) \cdot \color{blue}{h}\right) - g}{a + a}}\right) \]

    24. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot h\right) - g}{a + a}}\right) \]

    25. mul0-lft 95.7%

        \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

15. Applied rewrites 95.7%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt[3]{-2 \cdot a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{0}\right) - g}{a + a}}\right) \]
16. Add Preprocessing

Alternative 3: 95.7% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

FPCore

(FPCore (g h a)
  :precision binary64
  (fma
 (cbrt (/ -0.5 a))
 (cbrt (+ (+ g 0.0) g))
 (cbrt (/ (- (+ g 0.0) g) (+ a a)))))

C

double code(double g, double h, double a) {
	return fma(cbrt((-0.5 / a)), cbrt(((g + 0.0) + g)), cbrt((((g + 0.0) - g) / (a + a))));
}

Julia

function code(g, h, a)
	return fma(cbrt(Float64(-0.5 / a)), cbrt(Float64(Float64(g + 0.0) + g)), cbrt(Float64(Float64(Float64(g + 0.0) - g) / Float64(a + a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(g + 0.0), $MachinePrecision] + g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(N[(g + 0.0), $MachinePrecision] - g), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.3%

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

3. Applied rewrites 46.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right)} \]

4. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 32.2%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

6. Applied rewrites 32.2%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} + g}, \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}{a + a}}\right) \]

7. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 95.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

9. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) + g}, \sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-+.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. associate-*r/ N/A

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

    7. sum-to-mult-rev N/A

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

    8. *-commutative N/A

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

    9. lift-+.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. distribute-rgt1-in N/A

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

    12. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + \left(0 \cdot h\right) \cdot \frac{1}{2}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    13. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0 \cdot \frac{1}{2}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    15. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0 \cdot \color{blue}{h}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    16. metadata-eval N/A

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

    17. distribute-rgt1-in N/A

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

    18. lift-*.f64 N/A

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

    19. lift-+.f64 N/A

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

    20. lower-+.f64 95.9%

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + \color{blue}{\left(h + -1 \cdot h\right)}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    21. lift-+.f64 N/A

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

    22. lift-*.f64 N/A

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

    23. distribute-rgt1-in N/A

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

    24. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0 \cdot h\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

    25. mul0-lft 95.9%

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]

11. Applied rewrites 95.9%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + \color{blue}{0}\right) + g}, \sqrt[3]{\frac{g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right) - g}{a + a}}\right) \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{g \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right)} - g}{a + a}}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot \color{blue}{g} - g}{a + a}}\right) \]

    3. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    5. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{1}{2} \cdot \frac{h + -1 \cdot h}{g}\right) \cdot g - g}{a + a}}\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(1 + \frac{\frac{1}{2} \cdot \left(h + -1 \cdot h\right)}{g}\right) \cdot g - g}{a + a}}\right) \]

    7. sum-to-mult-rev N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{\frac{1}{2} \cdot \left(h + -1 \cdot h\right)}\right) - g}{a + a}}\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \color{blue}{\frac{1}{2}}\right) - g}{a + a}}\right) \]

    9. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    11. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(\left(-1 + 1\right) \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(0 \cdot h\right) \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    13. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot \frac{1}{2}\right) - g}{a + a}}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

    15. mul0-lft N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot \color{blue}{h}\right) - g}{a + a}}\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(-1 + 1\right) \cdot h\right) - g}{a + a}}\right) \]

    17. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    18. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot \color{blue}{h}\right)\right) - g}{a + a}}\right) \]

    19. lift-+.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    20. lower-+.f64 95.9%

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{\left(h + -1 \cdot h\right)}\right) - g}{a + a}}\right) \]

    21. lift-+.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + \color{blue}{-1 \cdot h}\right)\right) - g}{a + a}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(h + -1 \cdot \color{blue}{h}\right)\right) - g}{a + a}}\right) \]

    23. distribute-rgt1-in N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \left(-1 + 1\right) \cdot \color{blue}{h}\right) - g}{a + a}}\right) \]

    24. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{-1}{2}}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0 \cdot h\right) - g}{a + a}}\right) \]

    25. mul0-lft 95.9%

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + 0\right) - g}{a + a}}\right) \]

13. Applied rewrites 95.9%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{-0.5}{a}}, \sqrt[3]{\left(g + 0\right) + g}, \sqrt[3]{\frac{\left(g + \color{blue}{0}\right) - g}{a + a}}\right) \]
14. Add Preprocessing

Alternative 4: 74.0% accurate, 1.3× speedup

Math

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]

FPCore

(FPCore (g h a)
  :precision binary64
  (+
 (cbrt (* (/ 1.0 (* 2.0 a)) (* -0.5 (* h (/ h g)))))
 (cbrt (* -1.0 (/ g a)))))

C

double code(double g, double h, double a) {
	return cbrt(((1.0 / (2.0 * a)) * (-0.5 * (h * (h / g))))) + cbrt((-1.0 * (g / a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (-0.5 * (h * (h / g))))) + Math.cbrt((-1.0 * (g / a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(-0.5 * Float64(h * Float64(h / g))))) + cbrt(Float64(-1.0 * Float64(g / a))))
end

Wolfram

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

Derivation

1. Initial program 43.3%

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

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 26.8%

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

4. Applied rewrites 26.8%

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

5. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 70.2%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]

7. Applied rewrites 70.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-0.5 \cdot \frac{{h}^{2}}{g}\right)}} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 74.0%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{\color{blue}{g}}\right)\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]

9. Applied rewrites 74.0%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(-0.5 \cdot \left(h \cdot \color{blue}{\frac{h}{g}}\right)\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]
10. Add Preprocessing

Alternative 5: 72.5% accurate, 1.9× speedup

Math

\[\sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0}{a}} \]

FPCore

(FPCore (g h a)
  :precision binary64
  (- (cbrt (/ (- g) a)) (cbrt (/ 0.0 a))))

C

double code(double g, double h, double a) {
	return cbrt((-g / a)) - cbrt((0.0 / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((-g / a)) - Math.cbrt((0.0 / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-g) / a)) - cbrt(Float64(0.0 / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[N[(0.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.3%

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

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 26.8%

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

4. Applied rewrites 26.8%

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

6. Applied rewrites 26.9%

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

7. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\color{blue}{\frac{-1}{4} \cdot \frac{h + -1 \cdot h}{a}}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

      \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{-1}{4} \cdot \frac{h + -1 \cdot h}{a}} \]

   4. lower-*.f64 72.5%

      \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{-0.25 \cdot \frac{h + -1 \cdot h}{a}} \]

9. Applied rewrites 72.5%

   \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\color{blue}{-0.25 \cdot \frac{h + -1 \cdot h}{a}}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{h + -1 \cdot h}{a} \cdot \frac{-1}{4}} \]

    4. associate-*l/ N/A

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

    5. lift-+.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. distribute-rgt1-in N/A

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

    8. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{\left(0 \cdot h\right) \cdot \frac{-1}{4}}{a}} \]

    9. mul0-lft N/A

       \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0 \cdot \frac{-1}{4}}{a}} \]

    10. metadata-eval N/A

        \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0}{a}} \]

    11. mul0-lft N/A

        \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0 \cdot h}{a}} \]

    12. metadata-eval N/A

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

    13. distribute-rgt1-in N/A

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

    14. lift-*.f64 N/A

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

    15. lift-+.f64 N/A

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

    16. lift-/.f64 72.5%

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

    17. lift-+.f64 N/A

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

    18. lift-*.f64 N/A

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

    19. distribute-rgt1-in N/A

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

    20. metadata-eval N/A

        \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0 \cdot h}{a}} \]

    21. mul0-lft 72.5%

        \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\frac{0}{a}} \]

11. Applied rewrites 72.5%

    \[\leadsto \sqrt[3]{\frac{-g}{a}} - \sqrt[3]{\color{blue}{\frac{0}{a}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025214 
(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))))))))