2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.9%

Time: 6.8s

Alternatives: 4

Speedup: 3.9×

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: 44.1% 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: 95.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

2. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 73.3

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

4. Applied rewrites 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 95.9

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

6. Applied rewrites 95.9%

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

Alternative 2: 75.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 \cdot \sqrt[3]{\left(a \cdot a\right) \cdot g}}{a}\\ t_1 := \frac{1}{2 \cdot a}\\ t_2 := -\sqrt[3]{\frac{g}{a}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+71}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(g \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ (* -1.0 (cbrt (* (* a a) g))) a))
        (t_1 (/ 1.0 (* 2.0 a)))
        (t_2 (- (cbrt (/ g a)))))
   (if (<= t_1 -2e+159)
     t_2
     (if (<= t_1 -2e+34)
       t_0
       (if (<= t_1 1e+71)
         t_2
         (if (<= t_1 5e+132)
           t_0
           (* -1.0 (* g (cbrt (/ 1.0 (* a (* g g))))))))))))

C

double code(double g, double h, double a) {
	double t_0 = (-1.0 * cbrt(((a * a) * g))) / a;
	double t_1 = 1.0 / (2.0 * a);
	double t_2 = -cbrt((g / a));
	double tmp;
	if (t_1 <= -2e+159) {
		tmp = t_2;
	} else if (t_1 <= -2e+34) {
		tmp = t_0;
	} else if (t_1 <= 1e+71) {
		tmp = t_2;
	} else if (t_1 <= 5e+132) {
		tmp = t_0;
	} else {
		tmp = -1.0 * (g * cbrt((1.0 / (a * (g * g)))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = (-1.0 * Math.cbrt(((a * a) * g))) / a;
	double t_1 = 1.0 / (2.0 * a);
	double t_2 = -Math.cbrt((g / a));
	double tmp;
	if (t_1 <= -2e+159) {
		tmp = t_2;
	} else if (t_1 <= -2e+34) {
		tmp = t_0;
	} else if (t_1 <= 1e+71) {
		tmp = t_2;
	} else if (t_1 <= 5e+132) {
		tmp = t_0;
	} else {
		tmp = -1.0 * (g * Math.cbrt((1.0 / (a * (g * g)))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(Float64(-1.0 * cbrt(Float64(Float64(a * a) * g))) / a)
	t_1 = Float64(1.0 / Float64(2.0 * a))
	t_2 = Float64(-cbrt(Float64(g / a)))
	tmp = 0.0
	if (t_1 <= -2e+159)
		tmp = t_2;
	elseif (t_1 <= -2e+34)
		tmp = t_0;
	elseif (t_1 <= 1e+71)
		tmp = t_2;
	elseif (t_1 <= 5e+132)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(g * cbrt(Float64(1.0 / Float64(a * Float64(g * g))))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(N[(-1.0 * N[Power[N[(N[(a * a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision])}, If[LessEqual[t$95$1, -2e+159], t$95$2, If[LessEqual[t$95$1, -2e+34], t$95$0, If[LessEqual[t$95$1, 1e+71], t$95$2, If[LessEqual[t$95$1, 5e+132], t$95$0, N[(-1.0 * N[(g * N[Power[N[(1.0 / N[(a * N[(g * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e159 or -1.99999999999999989e34 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 1e71

   1. Initial program 44.1%

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

   2. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in g around 0

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -1.9999999999999999e159 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -1.99999999999999989e34 or 1e71 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 5.0000000000000001e132

   1. Initial program 44.1%

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

   2. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. cbrt-unprod N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. cbrt-unprod N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

   4. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt[3]{{a}^{2} \cdot g} \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{{a}^{2} \cdot {h}^{2}}{g}} \cdot \sqrt[3]{\frac{-1}{4}}}{a} \]

   9. Applied rewrites 36.9%

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

   10. Taylor expanded in g around -inf

       \[\leadsto \frac{-1 \cdot \sqrt[3]{{a}^{2} \cdot g}}{a} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 38.7

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

   12. Applied rewrites 38.7%

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

   if 5.0000000000000001e132 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.1%

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

   3. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{0.5}{a}\right)}^{0.3333333333333333}, {\left(\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right)}^{0.3333333333333333}, \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)\right)}\right)} \]

   4. Taylor expanded in g around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   6. Applied rewrites 37.9%

      \[\leadsto \color{blue}{-1 \cdot \left(g \cdot \left(\sqrt[3]{\frac{0}{a \cdot \left(g \cdot g\right)} \cdot 0.5} + \sqrt[3]{\frac{2}{a \cdot \left(g \cdot g\right)} \cdot 0.5}\right)\right)} \]

   7. Taylor expanded in g around 0

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

      1. lower-cbrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 38.2

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

   9. Applied rewrites 38.2%

      \[\leadsto -1 \cdot \left(g \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := -\sqrt[3]{\frac{g}{a}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1 \cdot \sqrt[3]{\left(a \cdot a\right) \cdot g}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (- (cbrt (/ g a)))))
   (if (<= t_0 -2e+159)
     t_1
     (if (<= t_0 -2e+34) (/ (* -1.0 (cbrt (* (* a a) g))) a) t_1))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = -cbrt((g / a));
	double tmp;
	if (t_0 <= -2e+159) {
		tmp = t_1;
	} else if (t_0 <= -2e+34) {
		tmp = (-1.0 * cbrt(((a * a) * g))) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = -Math.cbrt((g / a));
	double tmp;
	if (t_0 <= -2e+159) {
		tmp = t_1;
	} else if (t_0 <= -2e+34) {
		tmp = (-1.0 * Math.cbrt(((a * a) * g))) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = Float64(-cbrt(Float64(g / a)))
	tmp = 0.0
	if (t_0 <= -2e+159)
		tmp = t_1;
	elseif (t_0 <= -2e+34)
		tmp = Float64(Float64(-1.0 * cbrt(Float64(Float64(a * a) * g))) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision])}, If[LessEqual[t$95$0, -2e+159], t$95$1, If[LessEqual[t$95$0, -2e+34], N[(N[(-1.0 * N[Power[N[(N[(a * a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e159 or -1.99999999999999989e34 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.1%

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

   2. Taylor expanded in g around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in g around 0

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -1.9999999999999999e159 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -1.99999999999999989e34

   1. Initial program 44.1%

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

   2. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. cbrt-unprod N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. cbrt-unprod N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

   4. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt[3]{{a}^{2} \cdot g} \cdot \sqrt[3]{-1} + \sqrt[3]{\frac{{a}^{2} \cdot {h}^{2}}{g}} \cdot \sqrt[3]{\frac{-1}{4}}}{a} \]

   9. Applied rewrites 36.9%

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

   10. Taylor expanded in g around -inf

       \[\leadsto \frac{-1 \cdot \sqrt[3]{{a}^{2} \cdot g}}{a} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-cbrt.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 38.7

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

   12. Applied rewrites 38.7%

       \[\leadsto \frac{-1 \cdot \sqrt[3]{\left(a \cdot a\right) \cdot g}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

2. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 73.3

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

4. Applied rewrites 73.3%

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

5. Taylor expanded in g around 0

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 73.3

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

7. Applied rewrites 73.3%

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

Reproduce

herbie shell --seed 2025128 
(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))))))))