Ian Simplification

Percentage Accurate: 6.8% → 8.2%

Time: 5.8s

Alternatives: 7

Speedup: 1.1×

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: 6.8% 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: 8.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (- 0.5 (* 0.5 x))))))
   (*
    (+ 1.0 (* -0.5 (/ (* PI (- 1.0 (* 2.0 (/ t_0 PI)))) t_0)))
    (acos (sqrt (fma x -0.5 0.5))))))

C

double code(double x) {
	double t_0 = acos(sqrt((0.5 - (0.5 * x))));
	return (1.0 + (-0.5 * ((((double) M_PI) * (1.0 - (2.0 * (t_0 / ((double) M_PI))))) / t_0))) * acos(sqrt(fma(x, -0.5, 0.5)));
}

Julia

function code(x)
	t_0 = acos(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(pi * Float64(1.0 - Float64(2.0 * Float64(t_0 / pi)))) / t_0))) * acos(sqrt(fma(x, -0.5, 0.5))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 + N[(-0.5 * N[(N[(Pi * N[(1.0 - N[(2.0 * N[(t$95$0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $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 6.8%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

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

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 6.8%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. sub-to-mult N/A

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

   5. lift-PI.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

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

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

7. Applied rewrites 6.8%

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

8. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

10. Applied rewrites 8.2%

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

Alternative 2: 8.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* PI 0.5) PI)))
   (* (fma (/ (acos (sqrt (fma x -0.5 0.5))) t_0) 2.0 1.0) t_0)))

C

double code(double x) {
	double t_0 = (((double) M_PI) * 0.5) - ((double) M_PI);
	return fma((acos(sqrt(fma(x, -0.5, 0.5))) / t_0), 2.0, 1.0) * t_0;
}

Julia

function code(x)
	t_0 = Float64(Float64(pi * 0.5) - pi)
	return Float64(fma(Float64(acos(sqrt(fma(x, -0.5, 0.5))) / t_0), 2.0, 1.0) * t_0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] - Pi), $MachinePrecision]}, N[(N[(N[(N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

3. Applied rewrites 8.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. sub-to-mult N/A

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

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

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

5. Applied rewrites 8.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-to-mult-rev N/A

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

   5. sub-to-mult N/A

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

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

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

7. Applied rewrites 8.2%

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

Alternative 3: 8.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(0.5 * Pi), $MachinePrecision] - N[(N[ArcCos[N[Sqrt[N[(-0.5 * x + 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 8.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 8.2

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

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

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

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

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

   12. lower-fma.f64 N/A

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 8.2

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

5. Applied rewrites 8.2%

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

Alternative 4: 6.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0 + N[(0.5 * Pi), $MachinePrecision]), $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. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 6.8%

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

Alternative 5: 5.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Applied rewrites 8.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--r+ N/A

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

   5. lower--.f64 N/A

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

5. Applied rewrites 8.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.3%

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

Alternative 6: 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 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * -2.0 + N[(0.5 * Pi), $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. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 4.1

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 4.1

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

6. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025159 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))