2-ancestry mixing, positive discriminant

Percentage Accurate: 43.8% → 95.9%

Time: 9.7s

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: 43.8% 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, 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 45.0%

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

3. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
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 73.0

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

5. Applied rewrites 73.0%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g}{a}} \cdot 1} \]
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 95.7

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

7. Applied rewrites 95.7%

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

8. Final simplification 95.7%

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

Alternative 2: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+65} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+118}\right):\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt[3]{\left(a \cdot a\right) \cdot g}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))))
   (if (or (<= t_0 2e+65) (not (<= t_0 2e+118)))
     (- (cbrt (/ g a)))
     (/ (- (cbrt (* (* a a) g))) a))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double tmp;
	if ((t_0 <= 2e+65) || !(t_0 <= 2e+118)) {
		tmp = -cbrt((g / a));
	} else {
		tmp = -cbrt(((a * a) * g)) / a;
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double tmp;
	if ((t_0 <= 2e+65) || !(t_0 <= 2e+118)) {
		tmp = -Math.cbrt((g / a));
	} else {
		tmp = -Math.cbrt(((a * a) * g)) / a;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	tmp = 0.0
	if ((t_0 <= 2e+65) || !(t_0 <= 2e+118))
		tmp = Float64(-cbrt(Float64(g / a)));
	else
		tmp = Float64(Float64(-cbrt(Float64(Float64(a * a) * g))) / a);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e+65], N[Not[LessEqual[t$95$0, 2e+118]], $MachinePrecision]], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), N[((-N[Power[N[(N[(a * a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]) / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 2e65 or 1.99999999999999993e118 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   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 75.2

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

   8. Applied rewrites 75.2%

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

   if 2e65 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 1.99999999999999993e118

   1. Initial program 16.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. 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 16.4%

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

   6. Taylor expanded in g around -inf

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

   8. Applied rewrites 18.5%

      \[\leadsto \frac{-1 \cdot \left(g \cdot \mathsf{fma}\left(\sqrt[3]{\frac{\left(a \cdot a\right) \cdot 0}{g \cdot g}}, \sqrt[3]{0.5}, \sqrt[3]{\frac{\left(a \cdot a\right) \cdot -2}{g \cdot g} \cdot -0.5}\right)\right)}{a} \]

   9. Taylor expanded in g around 0

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

       1. lower-cbrt.f64 N/A

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

       2. lower-*.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 95.5

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

   11. Applied rewrites 95.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq 2 \cdot 10^{+65} \lor \neg \left(\frac{1}{2 \cdot a} \leq 2 \cdot 10^{+118}\right):\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt[3]{\left(a \cdot a\right) \cdot g}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% 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 45.0%

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

3. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
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 73.0

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

5. Applied rewrites 73.0%

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

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 73.0

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

8. Applied rewrites 73.0%

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

Reproduce

herbie shell --seed 2025060 
(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))))))))