2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 77.0%

Time: 14.4s

Alternatives: 2

Speedup: 2.0×

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: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := t_1 - g\\ t_3 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_2} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{2 \cdot a}}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5}{\frac{g}{h \cdot h}}} + t_0\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\sqrt[3]{t_2 \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (+ g g) (/ -0.5 a))))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (- t_1 g))
        (t_3
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_2))
          (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a)))))))
   (if (<= t_3 -5e-105)
     (+ (cbrt (* (/ 0.5 a) (/ -0.5 (/ g (* h h))))) t_0)
     (if (<= t_3 0.0)
       (+ (cbrt (* t_2 (/ 0.5 a))) (* (cbrt (/ -0.5 a)) (cbrt (+ g g))))
       (+ t_0 (cbrt (* (/ 0.5 a) (- g g))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g + g) * (-0.5 / a)));
	double t_1 = sqrt(((g * g) - (h * h)));
	double t_2 = t_1 - g;
	double t_3 = cbrt(((1.0 / (2.0 * a)) * t_2)) + cbrt(((g + t_1) * (-1.0 / (2.0 * a))));
	double tmp;
	if (t_3 <= -5e-105) {
		tmp = cbrt(((0.5 / a) * (-0.5 / (g / (h * h))))) + t_0;
	} else if (t_3 <= 0.0) {
		tmp = cbrt((t_2 * (0.5 / a))) + (cbrt((-0.5 / a)) * cbrt((g + g)));
	} else {
		tmp = t_0 + cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g + g) * (-0.5 / a)));
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	double t_2 = t_1 - g;
	double t_3 = Math.cbrt(((1.0 / (2.0 * a)) * t_2)) + Math.cbrt(((g + t_1) * (-1.0 / (2.0 * a))));
	double tmp;
	if (t_3 <= -5e-105) {
		tmp = Math.cbrt(((0.5 / a) * (-0.5 / (g / (h * h))))) + t_0;
	} else if (t_3 <= 0.0) {
		tmp = Math.cbrt((t_2 * (0.5 / a))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + g)));
	} else {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g - g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g + g) * Float64(-0.5 / a)))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	t_2 = Float64(t_1 - g)
	t_3 = Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * t_2)) + cbrt(Float64(Float64(g + t_1) * Float64(-1.0 / Float64(2.0 * a)))))
	tmp = 0.0
	if (t_3 <= -5e-105)
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 / Float64(g / Float64(h * h))))) + t_0);
	elseif (t_3 <= 0.0)
		tmp = Float64(cbrt(Float64(t_2 * Float64(0.5 / a))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + g))));
	else
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))));
	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]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - g), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(g + t$95$1), $MachinePrecision] * N[(-1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-105], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 / N[(g / N[(h * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Power[N[(t$95$2 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -4.99999999999999963e-105

   1. Initial program 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in g around inf 54.4%

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

   5. Taylor expanded in g around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. associate-/l* 91.3%

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

      3. unpow2 91.3%

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

   7. Simplified 91.3%

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

   if -4.99999999999999963e-105 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -0.0

   1. Initial program 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in g around inf 4.3%

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

      1. cbrt-prod 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

   8. Simplified 93.1%

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

   if -0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 32.3%

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

   3. Simplified 32.3%

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

   4. Taylor expanded in g around inf 19.1%

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

   5. Taylor expanded in g around inf 70.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \frac{-0.5}{\frac{g}{h \cdot h}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]

Alternative 2: 73.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g + g) * Float64(-0.5 / a))) + cbrt(Float64(Float64(0.5 / a) * Float64(g - g))))
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[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

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

3. Simplified 46.4%

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

4. Taylor expanded in g around inf 28.9%

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

5. Taylor expanded in g around inf 72.8%

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

6. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2023271 
(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))))))))