2-ancestry mixing, positive discriminant

Percentage Accurate: 44.7% → 95.8%

Time: 9.3s

Alternatives: 3

Speedup: 3.8×

Specification

Math

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

FPCore

(FPCore (g h a)
  :precision binary64
  (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
  (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

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

Initial Program: 44.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h a)
  :precision binary64
  (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
  (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

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

Alternative 1: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := g \cdot \left(1 + 0.5 \cdot \frac{h + -1 \cdot h}{g}\right)\\ \mathsf{fma}\left(\sqrt[3]{\frac{1}{a}}, \sqrt[3]{\frac{t\_0 + g}{-2}}, \sqrt[3]{\frac{t\_0 - g}{a + a}}\right) \end{array} \]

FPCore

(FPCore (g h a)
  :precision binary64
  (let* ((t_0 (* g (+ 1.0 (* 0.5 (/ (+ h (* -1.0 h)) g))))))
  (fma
   (cbrt (/ 1.0 a))
   (cbrt (/ (+ t_0 g) -2.0))
   (cbrt (/ (- t_0 g) (+ a a))))))

C

double code(double g, double h, double a) {
	double t_0 = g * (1.0 + (0.5 * ((h + (-1.0 * h)) / g)));
	return fma(cbrt((1.0 / a)), cbrt(((t_0 + g) / -2.0)), cbrt(((t_0 - g) / (a + a))));
}

Julia

function code(g, h, a)
	t_0 = Float64(g * Float64(1.0 + Float64(0.5 * Float64(Float64(h + Float64(-1.0 * h)) / g))))
	return fma(cbrt(Float64(1.0 / a)), cbrt(Float64(Float64(t_0 + g) / -2.0)), cbrt(Float64(Float64(t_0 - g) / Float64(a + a))))
end

Wolfram

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

Derivation

1. Initial program 44.7%

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

3. Applied rewrites 47.5%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 32.8%

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

6. Applied rewrites 32.8%

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

7. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 95.6%

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

9. Applied rewrites 95.6%

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

Alternative 2: 95.6% accurate, 2.2× speedup

Math

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

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 44.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. cbrt-prod N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. cbrt-prod N/A

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

3. Applied rewrites 23.9%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.8%

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

6. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\color{blue}{\sqrt[3]{a}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\color{blue}{\sqrt[3]{a}}} \]

   6. lower-neg.f64 95.8%

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

8. Applied rewrites 95.8%

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

Alternative 3: 73.7% accurate, 3.8× speedup

Math

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

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 44.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. cbrt-prod N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. cbrt-prod N/A

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

3. Applied rewrites 23.9%

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

4. Taylor expanded in g around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 95.8%

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

6. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 95.8%

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

   4. lift-/.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-undiv N/A

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

   8. lift-/.f64 N/A

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

   9. lower-cbrt.f64 73.7%

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

8. Applied rewrites 73.7%

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

Reproduce

herbie shell --seed 2025213 
(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))))))))