2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 3.9s

Alternatives: 5

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, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \frac{t\_0 \cdot t\_0 - 6.283185307179587 \cdot 6.283185307179587}{t\_0 - 6.283185307179587}, \pi \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (*
    2.0
    (sin
     (fma
      -0.3333333333333333
      (/
       (- (* t_0 t_0) (* 6.283185307179587 6.283185307179587))
       (- t_0 6.283185307179587))
      (* PI 0.5))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return 2.0 * sin(fma(-0.3333333333333333, (((t_0 * t_0) - (6.283185307179587 * 6.283185307179587)) / (t_0 - 6.283185307179587)), (((double) M_PI) * 0.5)));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return Float64(2.0 * sin(fma(-0.3333333333333333, Float64(Float64(Float64(t_0 * t_0) - Float64(6.283185307179587 * 6.283185307179587)) / Float64(t_0 - 6.283185307179587)), Float64(pi * 0.5))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[Sin[N[(-0.3333333333333333 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(6.283185307179587 * 6.283185307179587), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 6.283185307179587), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{6.283185307179587} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right)} \]

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

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{-1}{3}} \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{3}, \frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   12. add-flip N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   13. lower--.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{\frac{-7074237752028441}{1125899906842624}}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   15. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \frac{\color{blue}{\pi}}{2}\right)\right) \]

   16. mult-flip N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   18. lower-*.f64 98.5%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, \color{blue}{\pi \cdot 0.5}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, \pi \cdot 0.5\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}}, \pi \cdot \frac{1}{2}\right)\right) \]

   2. sub-flip N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) + \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   3. flip-+ N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}}, \pi \cdot \frac{1}{2}\right)\right) \]

   4. lower-unsound-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}}, \pi \cdot \frac{1}{2}\right)\right) \]

   5. lower-unsound--.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   6. lower-unsound-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right)} - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   7. lower-unsound-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{\frac{7074237752028441}{1125899906842624}} \cdot \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \frac{7074237752028441}{1125899906842624} \cdot \color{blue}{\frac{7074237752028441}{1125899906842624}}}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}, \pi \cdot \frac{1}{2}\right)\right) \]

   10. lower-unsound--.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \frac{7074237752028441}{1125899906842624} \cdot \frac{7074237752028441}{1125899906842624}}{\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}}, \pi \cdot \frac{1}{2}\right)\right) \]

   11. metadata-eval 99.9%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 6.283185307179587 \cdot 6.283185307179587}{\cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{6.283185307179587}}, \pi \cdot 0.5\right)\right) \]

8. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 6.283185307179587 \cdot 6.283185307179587}{\cos^{-1} \left(\frac{-g}{h}\right) - 6.283185307179587}}, \pi \cdot 0.5\right)\right) \]
9. Add Preprocessing

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, 1.5707963267948966\right)\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    -0.3333333333333333
    (- (acos (/ (- g) h)) -6.283185307179587)
    1.5707963267948966))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(-0.3333333333333333, (acos((-g / h)) - -6.283185307179587), 1.5707963267948966));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(-0.3333333333333333, Float64(acos(Float64(Float64(-g) / h)) - -6.283185307179587), 1.5707963267948966)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(-0.3333333333333333 * N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] - -6.283185307179587), $MachinePrecision] + 1.5707963267948966), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval 98.5%

      \[\leadsto 2 \cdot \cos \left(\left(\color{blue}{6.283185307179587} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right)} \]

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

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{-1}{3}} \cdot \left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{3}, \frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   12. add-flip N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   13. lower--.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{\frac{-7074237752028441}{1125899906842624}}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   15. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \frac{\color{blue}{\pi}}{2}\right)\right) \]

   16. mult-flip N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   18. lower-*.f64 98.5%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, \color{blue}{\pi \cdot 0.5}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, \pi \cdot 0.5\right)\right)} \]

7. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587, \color{blue}{1.5707963267948966}\right)\right) \]
8. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot -0.3333333333333333\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (* (- (acos (/ (- g) h)) -6.283185307179587) -0.3333333333333333))
  2.0))

C

double code(double g, double h) {
	return cos(((acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = cos(((acos((-g / h)) - (-6.283185307179587d0)) * (-0.3333333333333333d0))) * 2.0d0
end function

Java

public static double code(double g, double h) {
	return Math.cos(((Math.acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
}

Python

def code(g, h):
	return math.cos(((math.acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0

Julia

function code(g, h)
	return Float64(cos(Float64(Float64(acos(Float64(Float64(-g) / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = cos(((acos((-g / h)) - -6.283185307179587) * -0.3333333333333333)) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] - -6.283185307179587), $MachinePrecision] * -0.3333333333333333), $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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.4%

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

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

   1. lift-fma.f64 N/A

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

   2. metadata-eval N/A

      \[\leadsto \cos \left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{7074237752028441}{1125899906842624}}\right) \cdot 2 \]

   3. metadata-eval N/A

      \[\leadsto \cos \left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{-1}{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)}\right) \cdot 2 \]

   4. distribute-lft-in N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{3} \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) + \left(\mathsf{neg}\left(\frac{-7074237752028441}{1125899906842624}\right)\right)\right)\right)} \cdot 2 \]

   5. sub-flip N/A

      \[\leadsto \cos \left(\frac{-1}{3} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}\right)}\right) \cdot 2 \]

   6. lift--.f64 N/A

      \[\leadsto \cos \left(\frac{-1}{3} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}\right)}\right) \cdot 2 \]

   7. *-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - \frac{-7074237752028441}{1125899906842624}\right) \cdot \frac{-1}{3}\right)} \cdot 2 \]

   8. lower-*.f64 98.5%

      \[\leadsto \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot -0.3333333333333333\right)} \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot -0.3333333333333333\right)} \cdot 2 \]
7. Add Preprocessing

Alternative 4: 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. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{2.0943951023931957} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.4%

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

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

Alternative 5: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + 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.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. 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.f64 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%

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 97.6%

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

6. Applied rewrites 97.6%

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

Reproduce

herbie shell --seed 2025192 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))