2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 97.5%

Time: 12.1s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

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: 43.7% accurate, 1.0× speedup

Math

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

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: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g \cdot 2}}{\sqrt[3]{a}}, \sqrt[3]{-0.5}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 97.7%

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

11. Applied rewrites 98.0%

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

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{-g}{a}\\ t_2 := {\left(2 \cdot a\right)}^{-1}\\ t_3 := \sqrt[3]{t\_2 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ t_4 := \frac{h \cdot h}{g}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{t\_2 \cdot \left(t\_4 \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, t\_4, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (/ (- g) a))
        (t_2 (pow (* 2.0 a) -1.0))
        (t_3
         (+
          (cbrt (* t_2 (+ (- g) t_0)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0)))))
        (t_4 (/ (* h h) g)))
   (if (<= t_3 -5e-99)
     (+ (cbrt (* t_2 (* t_4 -0.5))) (cbrt (fma (/ 0.25 a) t_4 t_1)))
     (if (<= t_3 0.0)
       (+ (/ (cbrt (- (- g) g)) (cbrt (* 2.0 a))) (cbrt t_1))
       (* (cbrt -1.0) (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = -g / a;
	double t_2 = pow((2.0 * a), -1.0);
	double t_3 = cbrt((t_2 * (-g + t_0))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double t_4 = (h * h) / g;
	double tmp;
	if (t_3 <= -5e-99) {
		tmp = cbrt((t_2 * (t_4 * -0.5))) + cbrt(fma((0.25 / a), t_4, t_1));
	} else if (t_3 <= 0.0) {
		tmp = (cbrt((-g - g)) / cbrt((2.0 * a))) + cbrt(t_1);
	} else {
		tmp = cbrt(-1.0) * cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(-g) / a)
	t_2 = Float64(2.0 * a) ^ -1.0
	t_3 = Float64(cbrt(Float64(t_2 * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	t_4 = Float64(Float64(h * h) / g)
	tmp = 0.0
	if (t_3 <= -5e-99)
		tmp = Float64(cbrt(Float64(t_2 * Float64(t_4 * -0.5))) + cbrt(fma(Float64(0.25 / a), t_4, t_1)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(Float64(-g) - g)) / cbrt(Float64(2.0 * a))) + cbrt(t_1));
	else
		tmp = Float64(cbrt(-1.0) * cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-g) / a), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(t$95$2 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-99], N[(N[Power[N[(t$95$2 * N[(t$95$4 * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.25 / a), $MachinePrecision] * t$95$4 + t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.99999999999999969e-99

   1. Initial program 85.0%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in h around 0

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a} + \frac{1}{4} \cdot \frac{{h}^{2}}{a \cdot g}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

      12. lower-neg.f64 95.8

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]

   if -4.99999999999999969e-99 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. cbrt-div N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 85.9%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 37.9%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 78.3%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \color{blue}{\sqrt[3]{-1}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]

   8. Taylor expanded in g around inf

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

   10. Applied rewrites 77.4%

       \[\leadsto \sqrt[3]{-1} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{-g}{a}\\ t_2 := {\left(2 \cdot a\right)}^{-1}\\ t_3 := \sqrt[3]{t\_2 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ t_4 := \frac{h \cdot h}{g}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{t\_2 \cdot \left(t\_4 \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, t\_4, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 0.5}}{\sqrt[3]{a}} + \sqrt[3]{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (/ (- g) a))
        (t_2 (pow (* 2.0 a) -1.0))
        (t_3
         (+
          (cbrt (* t_2 (+ (- g) t_0)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0)))))
        (t_4 (/ (* h h) g)))
   (if (<= t_3 -5e-99)
     (+ (cbrt (* t_2 (* t_4 -0.5))) (cbrt (fma (/ 0.25 a) t_4 t_1)))
     (if (<= t_3 0.0)
       (+ (/ (cbrt (* (- (- g) g) 0.5)) (cbrt a)) (cbrt t_1))
       (* (cbrt -1.0) (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = -g / a;
	double t_2 = pow((2.0 * a), -1.0);
	double t_3 = cbrt((t_2 * (-g + t_0))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double t_4 = (h * h) / g;
	double tmp;
	if (t_3 <= -5e-99) {
		tmp = cbrt((t_2 * (t_4 * -0.5))) + cbrt(fma((0.25 / a), t_4, t_1));
	} else if (t_3 <= 0.0) {
		tmp = (cbrt(((-g - g) * 0.5)) / cbrt(a)) + cbrt(t_1);
	} else {
		tmp = cbrt(-1.0) * cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(-g) / a)
	t_2 = Float64(2.0 * a) ^ -1.0
	t_3 = Float64(cbrt(Float64(t_2 * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	t_4 = Float64(Float64(h * h) / g)
	tmp = 0.0
	if (t_3 <= -5e-99)
		tmp = Float64(cbrt(Float64(t_2 * Float64(t_4 * -0.5))) + cbrt(fma(Float64(0.25 / a), t_4, t_1)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(Float64(Float64(-g) - g) * 0.5)) / cbrt(a)) + cbrt(t_1));
	else
		tmp = Float64(cbrt(-1.0) * cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-g) / a), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(t$95$2 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-99], N[(N[Power[N[(t$95$2 * N[(t$95$4 * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.25 / a), $MachinePrecision] * t$95$4 + t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Power[N[(N[((-g) - g), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.99999999999999969e-99

   1. Initial program 85.0%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in h around 0

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a} + \frac{1}{4} \cdot \frac{{h}^{2}}{a \cdot g}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

      12. lower-neg.f64 95.8

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]

   if -4.99999999999999969e-99 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 4.4

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

   10. Applied rewrites 4.4%

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

       1. lift-cbrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. cbrt-div N/A

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

       6. lift-cbrt.f64 N/A

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

       7. lower-/.f64 N/A

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

   12. Applied rewrites 85.9%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 37.9%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 78.3%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \color{blue}{\sqrt[3]{-1}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]

   8. Taylor expanded in g around inf

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

   10. Applied rewrites 77.4%

       \[\leadsto \sqrt[3]{-1} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 0.5}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{-g}{a}\\ t_2 := {\left(2 \cdot a\right)}^{-1}\\ t_3 := \sqrt[3]{t\_2 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + t\_0\right)}\\ t_4 := \frac{h \cdot h}{g}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{t\_2 \cdot \left(t\_4 \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, t\_4, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (/ (- g) a))
        (t_2 (pow (* 2.0 a) -1.0))
        (t_3
         (+
          (cbrt (* t_2 (+ (- g) t_0)))
          (cbrt (* (/ -1.0 (* 2.0 a)) (+ g t_0)))))
        (t_4 (/ (* h h) g)))
   (if (<= t_3 -5e-99)
     (+ (cbrt (* t_2 (* t_4 -0.5))) (cbrt (fma (/ 0.25 a) t_4 t_1)))
     (if (<= t_3 0.0)
       (fma (cbrt (/ 0.5 a)) (cbrt (- (- g) g)) (cbrt t_1))
       (* (cbrt -1.0) (cbrt (/ g a)))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = -g / a;
	double t_2 = pow((2.0 * a), -1.0);
	double t_3 = cbrt((t_2 * (-g + t_0))) + cbrt(((-1.0 / (2.0 * a)) * (g + t_0)));
	double t_4 = (h * h) / g;
	double tmp;
	if (t_3 <= -5e-99) {
		tmp = cbrt((t_2 * (t_4 * -0.5))) + cbrt(fma((0.25 / a), t_4, t_1));
	} else if (t_3 <= 0.0) {
		tmp = fma(cbrt((0.5 / a)), cbrt((-g - g)), cbrt(t_1));
	} else {
		tmp = cbrt(-1.0) * cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(Float64(-g) / a)
	t_2 = Float64(2.0 * a) ^ -1.0
	t_3 = Float64(cbrt(Float64(t_2 * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-1.0 / Float64(2.0 * a)) * Float64(g + t_0))))
	t_4 = Float64(Float64(h * h) / g)
	tmp = 0.0
	if (t_3 <= -5e-99)
		tmp = Float64(cbrt(Float64(t_2 * Float64(t_4 * -0.5))) + cbrt(fma(Float64(0.25 / a), t_4, t_1)));
	elseif (t_3 <= 0.0)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(Float64(-g) - g)), cbrt(t_1));
	else
		tmp = Float64(cbrt(-1.0) * cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-g) / a), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(t$95$2 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(h * h), $MachinePrecision] / g), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-99], N[(N[Power[N[(t$95$2 * N[(t$95$4 * -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.25 / a), $MachinePrecision] * t$95$4 + t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.99999999999999969e-99

   1. Initial program 85.0%

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

   3. Taylor expanded in g around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in h around 0

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a} + \frac{1}{4} \cdot \frac{{h}^{2}}{a \cdot g}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{a}}, \frac{{h}^{2}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \color{blue}{\frac{{h}^{2}}{g}}, -1 \cdot \frac{g}{a}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{\color{blue}{h \cdot h}}{g}, -1 \cdot \frac{g}{a}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \color{blue}{\frac{-1 \cdot g}{a}}\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot \frac{-1}{2}\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{\frac{1}{4}}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}\right)} \]

      12. lower-neg.f64 95.8

          \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{\color{blue}{-g}}{a}\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}} \]

   if -4.99999999999999969e-99 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

   1. Initial program 4.4%

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

   3. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   8. Applied rewrites 4.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. lower-fma.f64 N/A

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

   10. Applied rewrites 85.6%

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

   if 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 37.9%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 78.3%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \color{blue}{\sqrt[3]{-1}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]

   8. Taylor expanded in g around inf

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

   10. Applied rewrites 77.4%

       \[\leadsto \sqrt[3]{-1} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\frac{h \cdot h}{g} \cdot -0.5\right)} + \sqrt[3]{\mathsf{fma}\left(\frac{0.25}{a}, \frac{h \cdot h}{g}, \frac{-g}{a}\right)}\\ \mathbf{elif}\;\sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{-1}{2 \cdot a} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{\frac{-g}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1} \cdot \sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}, \sqrt[3]{-0.5}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.25}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  (* (/ (cbrt g) (cbrt a)) (cbrt 2.0))
  (cbrt -0.5)
  (* (/ (cbrt (* (/ h g) h)) (cbrt a)) (cbrt -0.25))))

C

double code(double g, double h, double a) {
	return fma(((cbrt(g) / cbrt(a)) * cbrt(2.0)), cbrt(-0.5), ((cbrt(((h / g) * h)) / cbrt(a)) * cbrt(-0.25)));
}

Julia

function code(g, h, a)
	return fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(2.0)), cbrt(-0.5), Float64(Float64(cbrt(Float64(Float64(h / g) * h)) / cbrt(a)) * cbrt(-0.25)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision] + N[(N[(N[Power[N[(N[(h / g), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.25, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 97.7%

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

11. Applied rewrites 97.7%

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}, \sqrt[3]{-0.5}, \frac{\sqrt[3]{\frac{h}{g} \cdot h}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.25}\right) \]
12. Add Preprocessing

Alternative 6: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g \cdot 2}}{\sqrt[3]{a}}, \sqrt[3]{-0.5}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{-0.5}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 96.6%

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

Alternative 7: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}, \sqrt[3]{-0.5}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \sqrt[3]{-0.25}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma
  (* (/ (cbrt g) (cbrt a)) (cbrt 2.0))
  (cbrt -0.5)
  (* (cbrt (* (/ h g) (/ h a))) (cbrt -0.25))))

C

double code(double g, double h, double a) {
	return fma(((cbrt(g) / cbrt(a)) * cbrt(2.0)), cbrt(-0.5), (cbrt(((h / g) * (h / a))) * cbrt(-0.25)));
}

Julia

function code(g, h, a)
	return fma(Float64(Float64(cbrt(g) / cbrt(a)) * cbrt(2.0)), cbrt(-0.5), Float64(cbrt(Float64(Float64(h / g) * Float64(h / a))) * cbrt(-0.25)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-0.5, 1/3], $MachinePrecision] + N[(N[Power[N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-0.25, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 96.2%

   \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{2}, \sqrt[3]{-0.5}, \sqrt[3]{\frac{h}{g} \cdot \frac{h}{a}} \cdot \sqrt[3]{-0.25}\right) \]
10. Add Preprocessing

Alternative 8: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{\frac{h}{a} \cdot -0.25}, \sqrt[3]{\frac{h}{g}}, \sqrt[3]{\frac{-g}{a}}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma (cbrt (* (/ h a) -0.25)) (cbrt (/ h g)) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return fma(cbrt(((h / a) * -0.25)), cbrt((h / g)), cbrt((-g / a)));
}

Julia

function code(g, h, a)
	return fma(cbrt(Float64(Float64(h / a) * -0.25)), cbrt(Float64(h / g)), cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(h / a), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(h / g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 97.7%

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

11. Applied rewrites 81.6%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{h}{a} \cdot -0.25}, \color{blue}{\sqrt[3]{\frac{h}{g}}}, \sqrt[3]{\frac{-g}{a}}\right) \]
12. Add Preprocessing

Alternative 9: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g \cdot 1}, \sqrt[3]{0}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma (cbrt (/ 0.5 a)) (cbrt (- (- g) (* g 1.0))) (cbrt 0.0)))

C

double code(double g, double h, double a) {
	return fma(cbrt((0.5 / a)), cbrt((-g - (g * 1.0))), cbrt(0.0));
}

Julia

function code(g, h, a)
	return fma(cbrt(Float64(0.5 / a)), cbrt(Float64(Float64(-g) - Float64(g * 1.0))), cbrt(0.0))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[((-g) - N[(g * 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[0.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

4. Applied rewrites 14.4%

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

5. Taylor expanded in g around inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\color{blue}{\frac{1}{4} \cdot \frac{h + -1 \cdot h}{a}}}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \color{blue}{\frac{h + -1 \cdot h}{a}}}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot h}}{a}}\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + 1\right) \cdot h}{a}}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} + 1\right) \cdot h}{a}}\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\color{blue}{\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)} \cdot h}{a}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\color{blue}{\left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot h}}{a}}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\left(1 + \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right) \cdot h}{a}}\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{\left(1 + \color{blue}{-1}\right) \cdot h}{a}}\right) \]

   10. metadata-eval 30.6

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{0.25 \cdot \frac{\color{blue}{0} \cdot h}{a}}\right) \]

7. Applied rewrites 30.6%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}, \sqrt[3]{\color{blue}{0.25 \cdot \frac{0 \cdot h}{a}}}\right) \]

8. Taylor expanded in g around -inf

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \left(-1 \cdot g\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \left(-1 \cdot g\right) \cdot \color{blue}{-1}}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{\left(-1 \cdot g\right) \cdot -1}}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot -1}, \sqrt[3]{\frac{1}{4} \cdot \frac{0 \cdot h}{a}}\right) \]

   6. lower-neg.f64 96.0

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{\left(-g\right)} \cdot -1}, \sqrt[3]{0.25 \cdot \frac{0 \cdot h}{a}}\right) \]

10. Applied rewrites 96.0%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - \color{blue}{\left(-g\right) \cdot -1}}, \sqrt[3]{0.25 \cdot \frac{0 \cdot h}{a}}\right) \]

11. Final simplification 96.0%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g \cdot 1}, \sqrt[3]{0}\right) \]
12. Add Preprocessing

Alternative 10: 73.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 80.6%

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

7. Applied rewrites 81.6%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{g}{a}}, \color{blue}{\sqrt[3]{-1}}, \sqrt[3]{-0.25 \cdot \left(\frac{h}{a} \cdot \frac{h}{g}\right)}\right) \]

8. Taylor expanded in g around inf

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

10. Applied rewrites 79.8%

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

Reproduce

herbie shell --seed 2024342 
(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))))))))