2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 96.0%

Time: 8.8s

Alternatives: 3

Speedup: 2.6×

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: 43.4% 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: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.6%

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

3. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
4. 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 72.4

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

5. Applied rewrites 72.4%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g}{a}} \cdot 1} \]
6. 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

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

7. Applied rewrites 95.9%

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

8. Final simplification 95.9%

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

Alternative 2: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;\sqrt[3]{t\_1 \cdot \left(\left(-g\right) + t\_0\right)} + \sqrt[3]{\left(-t\_1\right) \cdot \left(g + t\_0\right)} \leq -5 \cdot 10^{+98}:\\ \;\;\;\;g \cdot \left(0 + \sqrt[3]{\frac{-2}{a \cdot \left(g \cdot g\right)} \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (sqrt (- (* g g) (* h h)))) (t_1 (/ 1.0 (* 2.0 a))))
   (if (<=
        (+ (cbrt (* t_1 (+ (- g) t_0))) (cbrt (* (- t_1) (+ g t_0))))
        -5e+98)
     (* g (+ 0.0 (cbrt (* (/ -2.0 (* a (* g g))) 0.5))))
     (- (cbrt (/ g a))))))

C

double code(double g, double h, double a) {
	double t_0 = sqrt(((g * g) - (h * h)));
	double t_1 = 1.0 / (2.0 * a);
	double tmp;
	if ((cbrt((t_1 * (-g + t_0))) + cbrt((-t_1 * (g + t_0)))) <= -5e+98) {
		tmp = g * (0.0 + cbrt(((-2.0 / (a * (g * g))) * 0.5)));
	} else {
		tmp = -cbrt((g / a));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	double t_1 = 1.0 / (2.0 * a);
	double tmp;
	if ((Math.cbrt((t_1 * (-g + t_0))) + Math.cbrt((-t_1 * (g + t_0)))) <= -5e+98) {
		tmp = g * (0.0 + Math.cbrt(((-2.0 / (a * (g * g))) * 0.5)));
	} else {
		tmp = -Math.cbrt((g / a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_1 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if (Float64(cbrt(Float64(t_1 * Float64(Float64(-g) + t_0))) + cbrt(Float64(Float64(-t_1) * Float64(g + t_0)))) <= -5e+98)
		tmp = Float64(g * Float64(0.0 + cbrt(Float64(Float64(-2.0 / Float64(a * Float64(g * g))) * 0.5))));
	else
		tmp = Float64(-cbrt(Float64(g / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(t$95$1 * N[((-g) + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-t$95$1) * N[(g + t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -5e+98], N[(g * N[(0.0 + N[Power[N[(N[(-2.0 / N[(a * N[(g * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.9999999999999998e98

   1. Initial program 21.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 21.3%

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

   9. Taylor expanded in g around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

   11. Applied rewrites 99.1%

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

   12. Taylor expanded in g around 0

       \[\leadsto g \cdot \left(0 + \sqrt[3]{\frac{-2}{a \cdot \left(g \cdot g\right)} \cdot \frac{1}{2}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 99.1%

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

   if -4.9999999999999998e98 < (+.f64 (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 41.4%

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

   3. Taylor expanded in g around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
   4. 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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in g around 0

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 74.7

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

   8. Applied rewrites 74.7%

      \[\leadsto -\sqrt[3]{\frac{g}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\left(-\frac{1}{2 \cdot a}\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)} \leq -5 \cdot 10^{+98}:\\ \;\;\;\;g \cdot \left(0 + \sqrt[3]{\frac{-2}{a \cdot \left(g \cdot g\right)} \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 2.6× 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 40.6%

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

3. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
4. 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 72.4

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

5. Applied rewrites 72.4%

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

6. Taylor expanded in g around 0

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 72.4

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

8. Applied rewrites 72.4%

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

Reproduce

herbie shell --seed 2025058 
(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))))))))