2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.7s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\frac{\mathsf{fma}\left(-3, \pi, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2\right)}{-6}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (sin (/ (fma -3.0 PI (* (fma PI 2.0 (acos (/ (- g) h))) 2.0)) -6.0))))

C

double code(double g, double h) {
	return 2.0 * sin((fma(-3.0, ((double) M_PI), (fma(((double) M_PI), 2.0, acos((-g / h))) * 2.0)) / -6.0));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(Float64(fma(-3.0, pi, Float64(fma(pi, 2.0, acos(Float64(Float64(-g) / h))) * 2.0)) / -6.0)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(-3.0 * Pi + N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / -6.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

4. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

   5. frac-add N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 + -3 \cdot \mathsf{PI}\left(\right)}{-3 \cdot 2}\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 + -3 \cdot \mathsf{PI}\left(\right)}{-3 \cdot 2}\right)} \]

   7. +-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\color{blue}{-3 \cdot \mathsf{PI}\left(\right) + \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2}}{-3 \cdot 2}\right) \]

   8. lower-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 100.0

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(-3, \pi, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2\right)}{\color{blue}{-6}}\right) \]

6. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(-3, \pi, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2\right)}{-6}\right)} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.16666666666666666\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma -0.3333333333333333 (acos (/ (- g) h)) (* PI -0.16666666666666666)))
  2.0))

C

double code(double g, double h) {
	return sin(fma(-0.3333333333333333, acos((-g / h)), (((double) M_PI) * -0.16666666666666666))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(pi * -0.16666666666666666))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(Pi * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

4. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

   5. frac-add N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 + -3 \cdot \mathsf{PI}\left(\right)}{-3 \cdot 2}\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2 + -3 \cdot \mathsf{PI}\left(\right)}{-3 \cdot 2}\right)} \]

   7. +-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\color{blue}{-3 \cdot \mathsf{PI}\left(\right) + \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2}}{-3 \cdot 2}\right) \]

   8. lower-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 100.0

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(-3, \pi, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2\right)}{\color{blue}{-6}}\right) \]

6. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(-3, \pi, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 2\right)}{-6}\right)} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.16666666666666666\right)\right) \cdot 2} \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (fma PI 0.6666666666666666 (* 0.3333333333333333 (acos (/ (- g) h)))))))

C

double code(double g, double h) {
	return 2.0 * cos(fma(((double) M_PI), 0.6666666666666666, (0.3333333333333333 * acos((-g / h)))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(fma(pi, 0.6666666666666666, Float64(0.3333333333333333 * acos(Float64(Float64(-g) / h))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(Pi * 0.6666666666666666 + N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.4

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \pi + \frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \pi + \frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \pi + \frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \pi + \frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \pi + \frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2}{3} \cdot \mathsf{PI}\left(\right) + \frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. mul-1-neg N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right)\right) \]

   14. distribute-frac-neg N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   15. lower-acos.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   16. lift-/.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   17. lift-neg.f64 N/A

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

   18. lower-*.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

7. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\pi, \color{blue}{0.6666666666666666}, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (fma 0.3333333333333333 (acos (/ (- g) h)) (* 0.6666666666666666 PI)))))

C

double code(double g, double h) {
	return 2.0 * cos(fma(0.3333333333333333, acos((-g / h)), (0.6666666666666666 * ((double) M_PI))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(0.6666666666666666 * pi))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.4

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))