Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 6.2s

Alternatives: 7

Speedup: 1.3×

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: 6.8% 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, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\\ t_1 := \cos^{-1} t\_0\\ \left(\frac{\frac{1}{\mathsf{fma}\left(\frac{\pi}{t\_1}, 0.5, -1\right)} - 1}{t\_1} \cdot \sin^{-1} t\_0\right) \cdot t\_1 \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (fma x -0.5 0.5))) (t_1 (acos t_0)))
  (*
   (* (/ (- (/ 1.0 (fma (/ PI t_1) 0.5 -1.0)) 1.0) t_1) (asin t_0))
   t_1)))

C

double code(double x) {
	double t_0 = sqrt(fma(x, -0.5, 0.5));
	double t_1 = acos(t_0);
	return ((((1.0 / fma((((double) M_PI) / t_1), 0.5, -1.0)) - 1.0) / t_1) * asin(t_0)) * t_1;
}

Julia

function code(x)
	t_0 = sqrt(fma(x, -0.5, 0.5))
	t_1 = acos(t_0)
	return Float64(Float64(Float64(Float64(Float64(1.0 / fma(Float64(pi / t_1), 0.5, -1.0)) - 1.0) / t_1) * asin(t_0)) * t_1)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[t$95$0], $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 / N[(N[(Pi / t$95$1), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[ArcSin[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. sub-to-mult N/A

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

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

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

3. Applied rewrites 6.9%

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

   1. lift-/.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-/.f64 8.2%

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

5. Applied rewrites 8.2%

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

7. Applied rewrites 8.2%

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

Alternative 2: 8.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. sub-to-mult N/A

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

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

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

3. Applied rewrites 6.9%

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

   1. lift-/.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. asin-acos N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-/.f64 8.2%

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

5. Applied rewrites 8.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-flip N/A

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

   6. lift-/.f64 N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. associate--l+ N/A

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

7. Applied rewrites 8.2%

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

Alternative 3: 8.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (-
  1.0
  (/ (fma (acos (sqrt (fma x -0.5 0.5))) -2.0 PI) 1.5707963267948966))
 1.5707963267948966))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

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 8.2%

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

Alternative 4: 8.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-sqrt.f64 N/A

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

   7. pow1/2 N/A

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

   8. lift-fma.f64 N/A

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

   9. add-flip N/A

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

   10. *-commutative N/A

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

   11. add-flip N/A

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

   12. lift-fma.f64 N/A

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

   13. pow1/2 N/A

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

   14. lift-sqrt.f64 N/A

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

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

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

   16. lower-fma.f64 N/A

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

6. Applied rewrites 8.3%

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

Alternative 5: 6.8% accurate, 1.3× 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 6.8%

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

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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 4.1%

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

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 6.8%

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

10. Applied rewrites 6.8%

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

Alternative 6: 4.1% accurate, 1.5× speedup

Math

\[\cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (- (acos (sqrt 0.5)) (asin (sqrt 0.5))))

C

double code(double x) {
	return acos(sqrt(0.5)) - asin(sqrt(0.5));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = acos(sqrt(0.5d0)) - asin(sqrt(0.5d0))
end function

Java

public static double code(double x) {
	return Math.acos(Math.sqrt(0.5)) - Math.asin(Math.sqrt(0.5));
}

Python

def code(x):
	return math.acos(math.sqrt(0.5)) - math.asin(math.sqrt(0.5))

Julia

function code(x)
	return Float64(acos(sqrt(0.5)) - asin(sqrt(0.5)))
end

MATLAB

function tmp = code(x)
	tmp = acos(sqrt(0.5)) - asin(sqrt(0.5));
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] - N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 4.1%

       \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right)} - \sin^{-1} \left(\sqrt{0.5}\right) \]

6. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{0.5}\right) - \sin^{-1} \left(\sqrt{0.5}\right)} \]
7. Add Preprocessing

Alternative 7: 4.1% accurate, 1.9× 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 6.8%

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

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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 4.1%

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

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 2025212 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))