2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 7.1s

Alternatives: 2

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out-- N/A

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

   7. distribute-lft-out-- N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

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

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

   13. distribute-rgt-neg-in N/A

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

   14. lift-*.f64 N/A

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

   15. remove-double-neg N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 100.0

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

   18. lift-*.f64 N/A

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* (sin (* (+ PI (asin (/ g h))) -0.3333333333333333)) 2.0))

C

double code(double g, double h) {
	return sin(((((double) M_PI) + asin((g / h))) * -0.3333333333333333)) * 2.0;
}

Java

public static double code(double g, double h) {
	return Math.sin(((Math.PI + Math.asin((g / h))) * -0.3333333333333333)) * 2.0;
}

Python

def code(g, h):
	return math.sin(((math.pi + math.asin((g / h))) * -0.3333333333333333)) * 2.0

Julia

function code(g, h)
	return Float64(sin(Float64(Float64(pi + asin(Float64(g / h))) * -0.3333333333333333)) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = sin(((pi + asin((g / h))) * -0.3333333333333333)) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[(Pi + N[ArcSin[N[(g / h), $MachinePrecision]], $MachinePrecision]), $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. 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

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin \left(\left(\pi + \sin^{-1} \left(\frac{g}{h}\right)\right) \cdot -0.3333333333333333\right) \cdot 2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025157 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))