Ian Simplification

Percentage Accurate: 7.1% → 8.6%

Time: 5.6s

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: 7.1% 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: 8.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - x}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(t\_0 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{t\_0}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 x))))
   (if (<= x -4e-310)
     (- (/ PI 2.0) (* 2.0 (asin (* t_0 (sqrt 0.5)))))
     (- (/ PI 2.0) (* 2.0 (asin (/ t_0 (sqrt 2.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 - x));
	double tmp;
	if (x <= -4e-310) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((t_0 * sqrt(0.5))));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((t_0 / sqrt(2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 - x));
	double tmp;
	if (x <= -4e-310) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((t_0 * Math.sqrt(0.5))));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((t_0 / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 - x))
	tmp = 0
	if x <= -4e-310:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((t_0 * math.sqrt(0.5))))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((t_0 / math.sqrt(2.0))))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 - x))
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(t_0 * sqrt(0.5)))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(t_0 / sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 - x));
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = (pi / 2.0) - (2.0 * asin((t_0 * sqrt(0.5))));
	else
		tmp = (pi / 2.0) - (2.0 * asin((t_0 / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4e-310], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(t$95$0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 8.7%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   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. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. sqrt-unprod N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-sqrt.f64 8.6

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

   3. Applied rewrites 8.6%

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

   if -3.999999999999988e-310 < x

   1. Initial program 5.6%

      \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   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 8.5

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

   3. Applied rewrites 8.5%

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

Alternative 2: 8.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \frac{\pi}{2}\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \color{blue}{\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)\right)} \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   2. lift-asin.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\color{blue}{\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)} \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \left(\sqrt{\color{blue}{\frac{-1}{2} \cdot x + \frac{1}{2}}}\right) \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \color{blue}{\left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)} \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   5. lift-asin.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) \cdot \color{blue}{\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)}\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{-1}{2} \cdot x + \frac{1}{2}}}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) \cdot \sin^{-1} \color{blue}{\left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)}\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   8. asin-acos-rev N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right)} \cdot \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\frac{\color{blue}{\pi}}{2} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   11. negate-sub2 N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\color{blue}{\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right)} \cdot \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   12. asin-acos-rev N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right)}\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   13. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right) \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   14. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right) \cdot \left(\frac{\color{blue}{\pi}}{2} - \cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right)\right)\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   15. negate-sub2 N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \left(\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)\right)}\right) \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

   16. sqr-neg N/A

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \color{blue}{\left(\left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\cos^{-1} \left(\sqrt{\frac{-1}{2} \cdot x + \frac{1}{2}}\right) - \frac{\pi}{2}\right)\right)} \cdot 4}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right), 2, \frac{\pi}{2}\right)} \]

5. Applied rewrites 8.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} - 4 \cdot {\left(\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right) - \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \sin^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}} \]

8. Applied rewrites 8.6%

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

Alternative 3: 8.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.6%

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 8.6%

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

Alternative 4: 7.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (* (sqrt (- 1.0 x)) (sqrt 0.5))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin((sqrt((1.0 - x)) * sqrt(0.5))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin((Math.sqrt((1.0 - x)) * Math.sqrt(0.5))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin((math.sqrt((1.0 - x)) * math.sqrt(0.5))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(sqrt(Float64(1.0 - x)) * sqrt(0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin((sqrt((1.0 - x)) * sqrt(0.5))));
end

Wolfram

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

Derivation

1. Initial program 7.1%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. *-lft-identity N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. sqrt-unprod N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lower-sqrt.f64 7.0

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

3. Applied rewrites 7.0%

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

Alternative 5: 7.0% 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 7.1%

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

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 7.1%

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

Alternative 6: 4.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

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. Add Preprocessing

Alternative 7: 4.1% accurate, 1.4× speedup

Math

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

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 7.1%

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

3. Applied rewrites 7.1%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 7.1

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

6. Applied rewrites 7.1%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 4.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))