Ian Simplification

Percentage Accurate: 6.9% → 8.3%

Time: 6.3s

Alternatives: 6

Speedup: 1.2×

Specification

Math

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.3% accurate, 0.8× speedup

Math

\[\left(\left(\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 1.5707963267948966 - \pi\right) \cdot 0.4052847345693511\right) \cdot 1.5707963267948966\right) \cdot 1.5707963267948966 \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (*
    (fma 2.0 (acos (sqrt (fma x -0.5 0.5))) (- 1.5707963267948966 PI))
    0.4052847345693511)
   1.5707963267948966)
  1.5707963267948966))

C

double code(double x) {
	return ((fma(2.0, acos(sqrt(fma(x, -0.5, 0.5))), (1.5707963267948966 - ((double) M_PI))) * 0.4052847345693511) * 1.5707963267948966) * 1.5707963267948966;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(2.0, acos(sqrt(fma(x, -0.5, 0.5))), Float64(1.5707963267948966 - pi)) * 0.4052847345693511) * 1.5707963267948966) * 1.5707963267948966)
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 * N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(1.5707963267948966 - Pi), $MachinePrecision]), $MachinePrecision] * 0.4052847345693511), $MachinePrecision] * 1.5707963267948966), $MachinePrecision] * 1.5707963267948966), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.3%

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

4. Evaluated real constant 8.3%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

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

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

6. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), \pi\right)}{1.5707963267948966}\right) \cdot 1.5707963267948966} \]

8. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 1.5707963267948966 - \pi\right) \cdot 0.4052847345693511\right) \cdot 1.5707963267948966\right)} \cdot 1.5707963267948966 \]
9. Add Preprocessing

Alternative 2: 8.3% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, 1.5707963267948966 - \pi\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (acos (sqrt (fma -0.5 x 0.5))) 2.0 (- 1.5707963267948966 PI)))

C

double code(double x) {
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, (1.5707963267948966 - ((double) M_PI)));
}

Julia

function code(x)
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, Float64(1.5707963267948966 - pi))
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(1.5707963267948966 - Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.3%

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

4. Evaluated real constant 8.3%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2} \]

   4. sub-negate-rev N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \left(\frac{884279719003555}{562949953421312} - \pi\right)\right)\right)} \]

   5. sub-negate-rev N/A

      \[\leadsto \mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \color{blue}{\left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)}\right)\right) \]

   6. add-flip-rev N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 + \left(\pi - \frac{884279719003555}{562949953421312}\right)\right)}\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   9. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)\right)} \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2} + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   12. sub-negate-rev N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right), 2, \frac{884279719003555}{562949953421312} - \pi\right)} \]

6. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, 1.5707963267948966 - \pi\right)} \]
7. Add Preprocessing

Alternative 3: 6.9% accurate, 1.2× speedup

Math

\[\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right), -2, 1.5707963267948966\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (asin (sqrt (* 0.5 (- 1.0 x)))) -2.0 1.5707963267948966))

C

double code(double x) {
	return fma(asin(sqrt((0.5 * (1.0 - x)))), -2.0, 1.5707963267948966);
}

Julia

function code(x)
	return fma(asin(sqrt(Float64(0.5 * Float64(1.0 - x)))), -2.0, 1.5707963267948966)
end

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[N[(0.5 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0 + 1.5707963267948966), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 4.1%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{0.5}}\right) \]

5. Evaluated real constant 4.1%

   \[\leadsto \color{blue}{1.5707963267948966} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

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

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} + \left(\mathsf{neg}\left(2\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right) + \frac{884279719003555}{562949953421312}} \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{-2} \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right) + \frac{884279719003555}{562949953421312} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot -2} + \frac{884279719003555}{562949953421312} \]

   7. lower-fma.f64 4.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, 1.5707963267948966\right)} \]

7. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, 1.5707963267948966\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right)}, -2, 1.5707963267948966\right) \]
9. Step-by-step derivation

   1. lower-asin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right), -2, \frac{884279719003555}{562949953421312}\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right), -2, \frac{884279719003555}{562949953421312}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\frac{1}{2} \cdot \left(1 - x\right)}\right), -2, \frac{884279719003555}{562949953421312}\right) \]

   4. lower--.f64 6.9%

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right), -2, 1.5707963267948966\right) \]

10. Applied rewrites 6.9%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right)}, -2, 1.5707963267948966\right) \]
11. Add Preprocessing

Alternative 4: 6.8% accurate, 1.3× speedup

Math

\[\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, -1.5707963267948968\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (acos (sqrt (fma -0.5 x 0.5))) 2.0 -1.5707963267948968))

C

double code(double x) {
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, -1.5707963267948968);
}

Julia

function code(x)
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, -1.5707963267948968)
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0 + -1.5707963267948968), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.3%

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

4. Evaluated real constant 8.3%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2} \]

   4. sub-negate-rev N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \left(\frac{884279719003555}{562949953421312} - \pi\right)\right)\right)} \]

   5. sub-negate-rev N/A

      \[\leadsto \mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \color{blue}{\left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)}\right)\right) \]

   6. add-flip-rev N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 + \left(\pi - \frac{884279719003555}{562949953421312}\right)\right)}\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   9. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)\right)} \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2} + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   12. sub-negate-rev N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right), 2, \frac{884279719003555}{562949953421312} - \pi\right)} \]

6. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, 1.5707963267948966 - \pi\right)} \]

7. Evaluated real constant 6.8%

   \[\leadsto \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \color{blue}{-1.5707963267948968}\right) \]
8. Add Preprocessing

Alternative 5: 5.4% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(\cos^{-1} \left(\sqrt{0.5}\right), 2, 1.5707963267948966 - \pi\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (acos (sqrt 0.5)) 2.0 (- 1.5707963267948966 PI)))

C

double code(double x) {
	return fma(acos(sqrt(0.5)), 2.0, (1.5707963267948966 - ((double) M_PI)));
}

Julia

function code(x)
	return fma(acos(sqrt(0.5)), 2.0, Float64(1.5707963267948966 - pi))
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(1.5707963267948966 - Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.3%

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

4. Evaluated real constant 8.3%

   \[\leadsto \color{blue}{1.5707963267948966} - \left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\frac{884279719003555}{562949953421312} - \pi\right) - \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2} \]

   4. sub-negate-rev N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \left(\frac{884279719003555}{562949953421312} - \pi\right)\right)\right)} \]

   5. sub-negate-rev N/A

      \[\leadsto \mathsf{neg}\left(\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 - \color{blue}{\left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)}\right)\right) \]

   6. add-flip-rev N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2 + \left(\pi - \frac{884279719003555}{562949953421312}\right)\right)}\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \cdot 2}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   9. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)\right)} \cdot 2\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2} + \left(\mathsf{neg}\left(\left(\pi - \frac{884279719003555}{562949953421312}\right)\right)\right) \]

   12. sub-negate-rev N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right), 2, \frac{884279719003555}{562949953421312} - \pi\right)} \]

6. Applied rewrites 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, 1.5707963267948966 - \pi\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right), 2, 1.5707963267948966 - \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 5.4%

   \[\leadsto \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\color{blue}{0.5}}\right), 2, 1.5707963267948966 - \pi\right) \]
10. Add Preprocessing

Alternative 6: 4.1% accurate, 1.8× speedup

Math

\[\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, 1.5707963267948966\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (asin (sqrt 0.5)) -2.0 1.5707963267948966))

C

double code(double x) {
	return fma(asin(sqrt(0.5)), -2.0, 1.5707963267948966);
}

Julia

function code(x)
	return fma(asin(sqrt(0.5)), -2.0, 1.5707963267948966)
end

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0 + 1.5707963267948966), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right) \]
3. Step-by-step derivation

4. Applied rewrites 4.1%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{0.5}}\right) \]

5. Evaluated real constant 4.1%

   \[\leadsto \color{blue}{1.5707963267948966} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{884279719003555}{562949953421312} - \color{blue}{2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

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

      \[\leadsto \color{blue}{\frac{884279719003555}{562949953421312} + \left(\mathsf{neg}\left(2\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right) + \frac{884279719003555}{562949953421312}} \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{-2} \cdot \sin^{-1} \left(\sqrt{\frac{1}{2}}\right) + \frac{884279719003555}{562949953421312} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1}{2}}\right) \cdot -2} + \frac{884279719003555}{562949953421312} \]

   7. lower-fma.f64 4.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, 1.5707963267948966\right)} \]

7. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, 1.5707963267948966\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))