2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 95.8%

Time: 22.6s

Alternatives: 8

Speedup: 2.1×

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: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in g around -inf 29.2%

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

   1. *-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in g around -inf 74.2%

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

   1. neg-mul-1 74.2%

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

10. Simplified 74.2%

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

    1. cbrt-prod 96.5%

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

12. Applied egg-rr 96.5%

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

13. Final simplification 96.5%

    \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \]
14. Add Preprocessing

Alternative 2: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -1.42 \cdot 10^{-83}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + -2 \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -1.42e-83)
     (+ t_0 (cbrt (/ 1.0 (/ a (- g)))))
     (if (<= a 5e-89)
       (+ (* (cbrt (/ 0.5 a)) (cbrt (* g -2.0))) (* -2.0 (cbrt (- g))))
       (+ t_0 (cbrt (/ (- g) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -1.42e-83) {
		tmp = t_0 + cbrt((1.0 / (a / -g)));
	} else if (a <= 5e-89) {
		tmp = (cbrt((0.5 / a)) * cbrt((g * -2.0))) + (-2.0 * cbrt(-g));
	} else {
		tmp = t_0 + cbrt((-g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -1.42e-83) {
		tmp = t_0 + Math.cbrt((1.0 / (a / -g)));
	} else if (a <= 5e-89) {
		tmp = (Math.cbrt((0.5 / a)) * Math.cbrt((g * -2.0))) + (-2.0 * Math.cbrt(-g));
	} else {
		tmp = t_0 + Math.cbrt((-g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -1.42e-83)
		tmp = Float64(t_0 + cbrt(Float64(1.0 / Float64(a / Float64(-g)))));
	elseif (a <= 5e-89)
		tmp = Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + Float64(-2.0 * cbrt(Float64(-g))));
	else
		tmp = Float64(t_0 + cbrt(Float64(Float64(-g) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -1.42e-83], N[(t$95$0 + N[Power[N[(1.0 / N[(a / (-g)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e-89], N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.4199999999999999e-83

   1. Initial program 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in g around -inf 39.1%

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

      1. *-commutative 39.1%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in g around -inf 93.0%

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

      1. neg-mul-1 93.0%

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

   10. Simplified 93.0%

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

       1. associate-*l/ 93.0%

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

       2. clear-num 93.0%

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

       3. *-commutative 93.0%

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

       4. associate-*r* 93.0%

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

       5. metadata-eval 93.0%

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

   12. Applied egg-rr 93.0%

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

   if -1.4199999999999999e-83 < a < 4.99999999999999967e-89

   1. Initial program 38.1%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in g around -inf 26.1%

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

      1. *-commutative 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in g around inf 11.1%

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

      1. cbrt-prod 95.5%

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

   10. Applied egg-rr 11.1%

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

       1. add-sqr-sqrt 5.6%

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

       2. sqrt-unprod 4.9%

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

       3. swap-sqr 6.1%

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

       4. count-2 6.1%

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

       5. count-2 6.1%

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

       6. swap-sqr 6.1%

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

       7. metadata-eval 6.1%

          \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

       8. metadata-eval 6.1%

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

       9. swap-sqr 6.1%

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

       10. *-commutative 6.1%

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

       11. *-commutative 6.1%

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

       12. frac-times 6.1%

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

       13. metadata-eval 6.1%

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

       14. metadata-eval 6.1%

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

       15. frac-times 6.1%

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

       16. swap-sqr 4.9%

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

       17. *-commutative 4.9%

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

       18. *-commutative 4.9%

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

   12. Applied egg-rr 11.1%

       \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \color{blue}{\sqrt[3]{g \cdot \frac{0.5}{a}} \cdot \sqrt[3]{-2}} \]

   14. Simplified 95.5%

       \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \color{blue}{-2 \cdot \sqrt[3]{-g}} \]

   if 4.99999999999999967e-89 < a

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in g around -inf 24.5%

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

      1. *-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in g around -inf 93.5%

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

      1. neg-mul-1 93.5%

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

   10. Simplified 93.5%

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

       1. expm1-log1p-u 68.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       2. expm1-udef 27.4%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       3. associate-*l/ 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       4. *-commutative 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       5. associate-*r* 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       6. metadata-eval 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-1} \cdot g}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       7. neg-mul-1 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

   12. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
   13. Step-by-step derivation

       1. expm1-def 68.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       2. expm1-log1p 93.6%

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

   14. Simplified 93.6%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-83}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + -2 \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in g around -inf 29.2%

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

   1. *-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in g around -inf 74.2%

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

   1. neg-mul-1 74.2%

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

10. Simplified 74.2%

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

    1. associate-*l/ 74.2%

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

    2. cbrt-div 96.5%

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

    3. *-commutative 96.5%

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

    4. associate-*r* 96.5%

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

    5. metadata-eval 96.5%

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

    6. neg-mul-1 96.5%

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

12. Applied egg-rr 96.5%

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

13. Final simplification 96.5%

    \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]
14. Add Preprocessing

Alternative 4: 89.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -9.8 \cdot 10^{-99}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -9.8e-99)
     (+ t_0 (cbrt (/ 1.0 (/ a (- g)))))
     (if (<= a 4.8e-89)
       (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -2.0))
       (+ t_0 (cbrt (/ (- g) a)))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -9.8e-99) {
		tmp = t_0 + cbrt((1.0 / (a / -g)));
	} else if (a <= 4.8e-89) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-2.0);
	} else {
		tmp = t_0 + cbrt((-g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -9.8e-99) {
		tmp = t_0 + Math.cbrt((1.0 / (a / -g)));
	} else if (a <= 4.8e-89) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-2.0);
	} else {
		tmp = t_0 + Math.cbrt((-g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -9.8e-99)
		tmp = Float64(t_0 + cbrt(Float64(1.0 / Float64(a / Float64(-g)))));
	elseif (a <= 4.8e-89)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-2.0));
	else
		tmp = Float64(t_0 + cbrt(Float64(Float64(-g) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -9.8e-99], N[(t$95$0 + N[Power[N[(1.0 / N[(a / (-g)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.8e-89], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.8000000000000006e-99

   1. Initial program 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in g around -inf 40.0%

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

      1. *-commutative 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in g around -inf 93.3%

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

      1. neg-mul-1 93.3%

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

   10. Simplified 93.3%

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

       1. associate-*l/ 93.3%

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

       2. clear-num 93.3%

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

       3. *-commutative 93.3%

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

       4. associate-*r* 93.3%

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

       5. metadata-eval 93.3%

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

   12. Applied egg-rr 93.3%

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

   if -9.8000000000000006e-99 < a < 4.80000000000000032e-89

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in g around -inf 25.0%

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

      1. *-commutative 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in g around inf 10.8%

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

      1. add-sqr-sqrt 5.6%

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

      2. sqrt-unprod 4.8%

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

      3. *-commutative 4.8%

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

      4. *-commutative 4.8%

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

      5. swap-sqr 5.2%

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

      6. *-commutative 5.2%

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

      7. *-commutative 5.2%

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

      8. swap-sqr 5.2%

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

      9. metadata-eval 5.2%

         \[\leadsto \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      10. metadata-eval 5.2%

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

      11. swap-sqr 5.2%

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

      12. count-2 5.2%

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

      13. count-2 5.2%

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

      14. frac-times 5.2%

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

      15. metadata-eval 5.2%

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

      16. metadata-eval 5.2%

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

      17. frac-times 5.2%

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

      18. swap-sqr 4.8%

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

      19. sqrt-unprod 5.6%

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

      20. add-sqr-sqrt 10.8%

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

      21. expm1-log1p-u 6.2%

          \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{0}{0 \cdot a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 41.3%

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

       1. add-sqr-sqrt 22.7%

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

       2. sqrt-unprod 12.0%

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

       3. swap-sqr 9.0%

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

       4. count-2 9.0%

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

       5. count-2 9.0%

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

       6. swap-sqr 9.0%

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

       7. metadata-eval 9.0%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

       8. metadata-eval 9.0%

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

       9. swap-sqr 9.0%

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

       10. *-commutative 9.0%

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

       11. *-commutative 9.0%

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

       12. frac-times 9.0%

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

       13. metadata-eval 9.0%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\sqrt{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.25}}{a \cdot a}}} \]

       14. metadata-eval 9.0%

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

       15. frac-times 9.0%

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

       16. swap-sqr 12.0%

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

       17. *-commutative 12.0%

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

       18. *-commutative 12.0%

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

       19. sqrt-unprod 22.7%

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

       20. add-sqr-sqrt 41.3%

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

   14. Applied egg-rr 92.5%

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

   16. Simplified 92.5%

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

   if 4.80000000000000032e-89 < a

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in g around -inf 24.5%

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

      1. *-commutative 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in g around -inf 93.5%

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

      1. neg-mul-1 93.5%

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

   10. Simplified 93.5%

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

       1. expm1-log1p-u 68.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       2. expm1-udef 27.4%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       3. associate-*l/ 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       4. *-commutative 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       5. associate-*r* 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       6. metadata-eval 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-1} \cdot g}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       7. neg-mul-1 27.4%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

   12. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
   13. Step-by-step derivation

       1. expm1-def 68.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

       2. expm1-log1p 93.6%

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

   14. Simplified 93.6%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}}} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-99}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -8.5 \cdot 10^{-65} \lor \neg \left(g \leq 2.6\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -8.5e-65) (not (<= g 2.6)))
   (+ (cbrt (/ (- g) a)) (cbrt -2.0))
   (+ (cbrt (- g)) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -8.5e-65) || !(g <= 2.6)) {
		tmp = cbrt((-g / a)) + cbrt(-2.0);
	} else {
		tmp = cbrt(-g) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -8.5e-65) || !(g <= 2.6)) {
		tmp = Math.cbrt((-g / a)) + Math.cbrt(-2.0);
	} else {
		tmp = Math.cbrt(-g) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -8.5e-65) || !(g <= 2.6))
		tmp = Float64(cbrt(Float64(Float64(-g) / a)) + cbrt(-2.0));
	else
		tmp = Float64(cbrt(Float64(-g)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -8.5e-65], N[Not[LessEqual[g, 2.6]], $MachinePrecision]], N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -8.5000000000000003e-65 or 2.60000000000000009 < g

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in g around -inf 28.6%

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

      1. *-commutative 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in g around inf 15.0%

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

      1. add-sqr-sqrt 6.9%

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

      2. sqrt-unprod 11.6%

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

      3. *-commutative 11.6%

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

      4. *-commutative 11.6%

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

      5. swap-sqr 17.3%

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

      6. *-commutative 17.3%

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

      7. *-commutative 17.3%

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

      8. swap-sqr 17.3%

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

      9. metadata-eval 17.3%

         \[\leadsto \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      10. metadata-eval 17.3%

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

      11. swap-sqr 17.3%

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

      12. count-2 17.3%

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

      13. count-2 17.3%

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

      14. frac-times 18.1%

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

      15. metadata-eval 18.1%

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

      16. metadata-eval 18.1%

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

      17. frac-times 17.3%

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

      18. swap-sqr 11.6%

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

      19. sqrt-unprod 6.9%

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

      20. add-sqr-sqrt 15.0%

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

      21. expm1-log1p-u 10.6%

          \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{0}{0 \cdot a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 44.9%

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

   13. Taylor expanded in g around 0 44.9%

       \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   14. Step-by-step derivation

       1. neg-mul-1 44.9%

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

       2. distribute-neg-frac 44.9%

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

   15. Simplified 44.9%

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

   if -8.5000000000000003e-65 < g < 2.60000000000000009

   1. Initial program 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in g around -inf 31.9%

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

      1. *-commutative 31.9%

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

   7. Simplified 31.9%

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

   8. Taylor expanded in g around inf 16.5%

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

   9. Taylor expanded in a around 0 16.5%

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

   11. Simplified 50.9%

       \[\leadsto \sqrt[3]{\color{blue}{-g}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -8.5 \cdot 10^{-65} \lor \neg \left(g \leq 2.6\right):\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in g around -inf 29.2%

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

   1. *-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in g around -inf 74.2%

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

   1. neg-mul-1 74.2%

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

10. Simplified 74.2%

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

    1. expm1-log1p-u 52.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    2. expm1-udef 26.4%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    3. associate-*l/ 26.4%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    4. *-commutative 26.4%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    5. associate-*r* 26.4%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    6. metadata-eval 26.4%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-1} \cdot g}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    7. neg-mul-1 26.4%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{-g}}{a}}\right)} - 1\right) + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

12. Applied egg-rr 26.4%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)} - 1\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]
13. Step-by-step derivation

    1. expm1-def 52.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} + \sqrt[3]{\left(g + \left(-g\right)\right) \cdot \frac{-0.5}{a}} \]

    2. expm1-log1p 74.2%

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

14. Simplified 74.2%

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

15. Final simplification 74.2%

    \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \]
16. Add Preprocessing

Alternative 7: 45.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in g around -inf 29.2%

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

   1. *-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in g around inf 15.3%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-unprod 14.1%

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

   3. *-commutative 14.1%

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

   4. *-commutative 14.1%

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

   5. swap-sqr 19.4%

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

   6. *-commutative 19.4%

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

   7. *-commutative 19.4%

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

   8. swap-sqr 19.4%

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

   9. metadata-eval 19.4%

      \[\leadsto \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. metadata-eval 19.4%

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

   11. swap-sqr 19.4%

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

   12. count-2 19.4%

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

   13. count-2 19.4%

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

   14. frac-times 20.1%

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

   15. metadata-eval 20.1%

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

   16. metadata-eval 20.1%

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

   17. frac-times 19.4%

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

   18. swap-sqr 14.1%

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

   19. sqrt-unprod 6.9%

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

   20. add-sqr-sqrt 15.3%

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

   21. expm1-log1p-u 11.0%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{0}{0 \cdot a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.7%

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

13. Taylor expanded in g around 0 42.7%

    \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
14. Step-by-step derivation

    1. neg-mul-1 42.7%

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

    2. distribute-neg-frac 42.7%

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

15. Simplified 42.7%

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

16. Final simplification 42.7%

    \[\leadsto \sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-2} \]
17. Add Preprocessing

Alternative 8: 4.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-2} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (cbrt -2.0))

C

double code(double g, double h, double a) {
	return cbrt(-2.0);
}

Java

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

Julia

function code(g, h, a)
	return cbrt(-2.0)
end

Wolfram

code[g_, h_, a_] := N[Power[-2.0, 1/3], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in g around -inf 29.2%

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

   1. *-commutative 29.2%

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

7. Simplified 29.2%

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

8. Taylor expanded in g around inf 15.3%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-unprod 14.1%

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

   3. *-commutative 14.1%

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

   4. *-commutative 14.1%

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

   5. swap-sqr 19.4%

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

   6. *-commutative 19.4%

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

   7. *-commutative 19.4%

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

   8. swap-sqr 19.4%

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

   9. metadata-eval 19.4%

      \[\leadsto \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. metadata-eval 19.4%

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

   11. swap-sqr 19.4%

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

   12. count-2 19.4%

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

   13. count-2 19.4%

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

   14. frac-times 20.1%

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

   15. metadata-eval 20.1%

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

   16. metadata-eval 20.1%

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

   17. frac-times 19.4%

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

   18. swap-sqr 14.1%

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

   19. sqrt-unprod 6.9%

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

   20. add-sqr-sqrt 15.3%

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

   21. expm1-log1p-u 11.0%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)\right)}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\frac{0}{0 \cdot a}\right)} - 1}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 42.7%

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

13. Taylor expanded in g around 0 4.6%

    \[\leadsto \color{blue}{\sqrt[3]{-2}} \]

14. Final simplification 4.6%

    \[\leadsto \sqrt[3]{-2} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))))