Ian Simplification

Percentage Accurate: 7.3% → 8.7%

Time: 5.8s

Alternatives: 7

Speedup: 0.9×

Specification

Math

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

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: 7.3% accurate, 1.0× speedup

Math

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

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: 8.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \pi, -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.66e-162)
   (fma -2.0 (asin (sqrt (fma -0.5 x 0.5))) (* 0.5 PI))
   (fma 0.5 PI (* -2.0 (- (* 0.5 PI) (acos (sqrt 0.5)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.66e-162) {
		tmp = fma(-2.0, asin(sqrt(fma(-0.5, x, 0.5))), (0.5 * ((double) M_PI)));
	} else {
		tmp = fma(0.5, ((double) M_PI), (-2.0 * ((0.5 * ((double) M_PI)) - acos(sqrt(0.5)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.66e-162)
		tmp = fma(-2.0, asin(sqrt(fma(-0.5, x, 0.5))), Float64(0.5 * pi));
	else
		tmp = fma(0.5, pi, Float64(-2.0 * Float64(Float64(0.5 * pi) - acos(sqrt(0.5)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.66e-162], N[(-2.0 * N[ArcSin[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision], N[(0.5 * Pi + N[(-2.0 * N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.66e-162

   1. Initial program 13.3%

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

   2. Taylor expanded in x around 0

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

      1. negate-sub N/A

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

      2. +-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 13.3%

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

   if -1.66e-162 < x

   1. Initial program 5.0%

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

      1. lift-asin.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. sqrt-unprod N/A

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

      11. asin-acos N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. negate-sub N/A

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

      17. mul-1-neg N/A

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

      18. metadata-eval N/A

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

      19. mul-1-neg N/A

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

      20. negate-sub N/A

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

   3. Applied rewrites 8.0%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lift-PI.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \]

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

         \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right) \]

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 8.0%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-sqrt.f64 5.9

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

   9. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(0.5, \pi, -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 8.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{1}{{\left(\mathsf{fma}\left(x, -0.5, 0.5\right)\right)}^{-0.5}}\right)\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (* (* PI 0.5) (* PI 0.5))
   (pow (* (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 2.0) 2.0))
  (fma
   PI
   0.5
   (* (- (* PI 0.5) (acos (/ 1.0 (pow (fma x -0.5 0.5) -0.5)))) 2.0))))

C

double code(double x) {
	return (((((double) M_PI) * 0.5) * (((double) M_PI) * 0.5)) - pow((((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) * 2.0), 2.0)) / fma(((double) M_PI), 0.5, (((((double) M_PI) * 0.5) - acos((1.0 / pow(fma(x, -0.5, 0.5), -0.5)))) * 2.0));
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(pi * 0.5) * Float64(pi * 0.5)) - (Float64(Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) * 2.0) ^ 2.0)) / fma(pi, 0.5, Float64(Float64(Float64(pi * 0.5) - acos(Float64(1.0 / (fma(x, -0.5, 0.5) ^ -0.5)))) * 2.0)))
end

Wolfram

code[x_] := N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi * 0.5 + N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(1.0 / N[Power[N[(x * -0.5 + 0.5), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. sqrt-unprod N/A

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

   11. asin-acos N/A

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

   12. lift-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower--.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. negate-sub N/A

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

   17. mul-1-neg N/A

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

   18. metadata-eval N/A

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

   19. mul-1-neg N/A

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

   20. negate-sub N/A

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

3. Applied rewrites 8.7%

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

4. Taylor expanded in x around 0

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

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

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

   2. lift-PI.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \]

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

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right) \]

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 8.7%

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

8. Applied rewrites 8.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 8.7

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

10. Applied rewrites 8.7%

    \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{1}{{\left(\mathsf{fma}\left(x, -0.5, 0.5\right)\right)}^{-0.5}}\right)\right) \cdot 2\right)} \]
11. Add Preprocessing

Alternative 3: 8.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ \frac{\mathsf{fma}\left(\pi \cdot \pi, 0.25, -4 \cdot {t\_0}^{2}\right)}{\mathsf{fma}\left(t\_0, 2, 0.5 \cdot \pi\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* 0.5 PI) (acos (sqrt (fma x -0.5 0.5))))))
   (/ (fma (* PI PI) 0.25 (* -4.0 (pow t_0 2.0))) (fma t_0 2.0 (* 0.5 PI)))))

C

double code(double x) {
	double t_0 = (0.5 * ((double) M_PI)) - acos(sqrt(fma(x, -0.5, 0.5)));
	return fma((((double) M_PI) * ((double) M_PI)), 0.25, (-4.0 * pow(t_0, 2.0))) / fma(t_0, 2.0, (0.5 * ((double) M_PI)));
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 * pi) - acos(sqrt(fma(x, -0.5, 0.5))))
	return Float64(fma(Float64(pi * pi), 0.25, Float64(-4.0 * (t_0 ^ 2.0))) / fma(t_0, 2.0, Float64(0.5 * pi)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25 + N[(-4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 2.0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.3%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. sqrt-unprod N/A

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

   11. asin-acos N/A

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

   12. lift-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower--.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. negate-sub N/A

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

   17. mul-1-neg N/A

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

   18. metadata-eval N/A

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

   19. mul-1-neg N/A

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

   20. negate-sub N/A

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

3. Applied rewrites 8.7%

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

4. Taylor expanded in x around 0

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

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

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

   2. lift-PI.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \]

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

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right) \]

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 8.7%

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

8. Applied rewrites 8.7%

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

10. Applied rewrites 8.7%

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, 0.25, -4 \cdot {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}\right)}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 2, 0.5 \cdot \pi\right)}} \]
11. Add Preprocessing

Alternative 4: 7.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, \pi, -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

   1. lift-asin.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. sqrt-unprod N/A

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

   11. asin-acos N/A

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

   12. lift-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower--.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. negate-sub N/A

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

   17. mul-1-neg N/A

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

   18. metadata-eval N/A

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

   19. mul-1-neg N/A

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

   20. negate-sub N/A

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

3. Applied rewrites 8.7%

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

4. Taylor expanded in x around 0

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

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

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

   2. lift-PI.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right) \]

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

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \pi + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \cos^{-1} \left(\sqrt{\frac{1}{2} - \frac{1}{2} \cdot x}\right)\right) \]

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 8.7%

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

Alternative 5: 7.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{0.5}\right), 0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fma -2.0 (asin (sqrt 0.5)) (* 0.5 PI))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(-2.0, asin(sqrt(0.5)), (0.5 * ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = fma(-2.0, asin(sqrt(0.5)), Float64(0.5 * pi));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(-2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 9.0%

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

   2. Taylor expanded in x around 0

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

      1. negate-sub N/A

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

      2. +-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{\frac{1}{2}}\right), \frac{1}{2} \cdot \pi\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 6.0%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-sqrt.f64 8.4

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

   3. Applied rewrites 8.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 5.8%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{\color{blue}{1}}{\sqrt{2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 5.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 0.5 \cdot \pi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in x around 0

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

   1. negate-sub N/A

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

   2. +-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 7.3%

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

Alternative 7: 4.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{0.5}\right), 0.5 \cdot \pi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in x around 0

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

   1. negate-sub N/A

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

   2. +-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 7.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{\frac{1}{2}}\right), \frac{1}{2} \cdot \pi\right) \]
6. Step-by-step derivation

7. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2025118 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))