2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 4.7s

Alternatives: 4

Speedup: 1.1×

Specification

Math

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

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

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

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: 99.9% accurate, 1.0× speedup

Math

\[\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma
    (fma PI 2.0 (acos (/ (- g) h)))
    -0.3333333333333333
    1.5707963267948966))
  2.0))

C

double code(double g, double h) {
	return sin(fma(fma(((double) M_PI), 2.0, acos((-g / h))), -0.3333333333333333, 1.5707963267948966)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(fma(pi, 2.0, acos(Float64(Float64(-g) / h))), -0.3333333333333333, 1.5707963267948966)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.5707963267948966), $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. Step-by-step derivation

   1. lift-cos.64 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.64 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

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

3. Applied rewrites 98.5%

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

4. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \color{blue}{1.5707963267948966}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right) \cdot 2} \]

   3. lower-*.f64 98.5%

      \[\leadsto \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + 1.5707963267948966\right) \cdot 2} \]

   4. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right)} \cdot 2 \]

   5. lift-/.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   6. mult-flip N/A

      \[\leadsto \sin \left(\color{blue}{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{-3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   7. metadata-eval N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{\frac{-1}{3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   8. lower-fma.f64 99.9%

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right)} \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right) \cdot 2} \]
7. Add Preprocessing

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\mathsf{fma}\left(0.3333333333333333, \pi - \cos^{-1} \left(\frac{g}{h}\right), 2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma 0.3333333333333333 (- PI (acos (/ g h))) 2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(fma(0.3333333333333333, (((double) M_PI) - acos((g / h))), 2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(0.3333333333333333, Float64(pi - acos(Float64(g / h))), 2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(0.3333333333333333 * N[(Pi - N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 2.0943951023931957), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.4%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{2.0943951023931957}\right)\right) \cdot 2 \]
5. Step-by-step derivation

   1. lift-acos.64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\frac{-g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   3. lift-neg.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right), \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   4. distribute-frac-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   5. acos-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   6. lift-PI.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\pi} - \cos^{-1} \left(\frac{g}{h}\right), \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   7. lower--.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   8. lower-acos.64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \pi - \color{blue}{\cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   9. lower-/.f64 98.5%

      \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \pi - \cos^{-1} \color{blue}{\left(\frac{g}{h}\right)}, 2.0943951023931957\right)\right) \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}, 2.0943951023931957\right)\right) \cdot 2 \]
7. Add Preprocessing

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (fma 0.3333333333333333 (acos (/ (- g) h)) 2.0943951023931957)) 2.0))

C

double code(double g, double h) {
	return cos(fma(0.3333333333333333, acos((-g / h)), 2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), 2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + 2.0943951023931957), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.4%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{2.0943951023931957}\right)\right) \cdot 2 \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.2× speedup

Math

\[\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (sin (fma 0.3333333333333333 (acos (/ (- g) h)) 3.6651914291880923)) 2.0))

C

double code(double g, double h) {
	return sin(fma(0.3333333333333333, acos((-g / h)), 3.6651914291880923)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), 3.6651914291880923)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + 3.6651914291880923), $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. Step-by-step derivation

   1. lift-cos.64 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. sin-+PI/2-rev N/A

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

   3. lower-sin.64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

3. Applied rewrites 97.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{fma}\left(\pi, 0.6666666666666666, \pi \cdot 0.5\right)\right)\right)} \]

4. Evaluated real constant 97.6%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{3.6651914291880923}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right) \cdot 2} \]

   3. lower-*.f64 97.6%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2} \]

6. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))