2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 6.0s

Alternatives: 2

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\frac{\mathsf{fma}\left(2 \cdot {\pi}^{0.5}, {\pi}^{0.5}, \cos^{-1} \left(\frac{-1 \cdot g}{h}\right)\right)}{-3} + {\pi}^{0.5} \cdot \frac{{\pi}^{0.5}}{2}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (+
    (/ (fma (* 2.0 (pow PI 0.5)) (pow PI 0.5) (acos (/ (* -1.0 g) h))) -3.0)
    (* (pow PI 0.5) (/ (pow PI 0.5) 2.0))))))

C

double code(double g, double h) {
	return 2.0 * sin(((fma((2.0 * pow(((double) M_PI), 0.5)), pow(((double) M_PI), 0.5), acos(((-1.0 * g) / h))) / -3.0) + (pow(((double) M_PI), 0.5) * (pow(((double) M_PI), 0.5) / 2.0))));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(Float64(Float64(fma(Float64(2.0 * (pi ^ 0.5)), (pi ^ 0.5), acos(Float64(Float64(-1.0 * g) / h))) / -3.0) + Float64((pi ^ 0.5) * Float64((pi ^ 0.5) / 2.0)))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(N[(N[(2.0 * N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.5], $MachinePrecision] + N[ArcCos[N[(N[(-1.0 * g), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision] + N[(N[Power[Pi, 0.5], $MachinePrecision] * N[(N[Power[Pi, 0.5], $MachinePrecision] / 2.0), $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. 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%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. unpow1/2 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. unpow1/2 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (* -1.0 g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos(((-1.0 * 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(((-1.0 * g) / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos(((-1.0 * 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(-1.0 * g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos(((-1.0 * 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[(N[(-1.0 * g), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] / 3.0), $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. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2025066 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))