Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 4.8s

Alternatives: 5

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: 8.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\pi \cdot \pi\right)\\ t_1 := \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\\ t_2 := t\_1 \cdot \left(t\_1 - 0.5 \cdot \pi\right)\\ \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {t\_1}^{3}\right)}{\frac{{t\_0}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(t\_0, t\_0, t\_2 \cdot t\_2 - t\_0 \cdot t\_2\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.25 (* PI PI)))
        (t_1 (* (- (* 0.5 PI) (acos (sqrt (* (- 1.0 x) 0.5)))) -2.0))
        (t_2 (* t_1 (- t_1 (* 0.5 PI)))))
   (/
    (fma (pow PI 3.0) 0.125 (pow t_1 3.0))
    (/
     (+ (pow t_0 3.0) (pow t_2 3.0))
     (fma t_0 t_0 (- (* t_2 t_2) (* t_0 t_2)))))))

C

double code(double x) {
	double t_0 = 0.25 * (((double) M_PI) * ((double) M_PI));
	double t_1 = ((0.5 * ((double) M_PI)) - acos(sqrt(((1.0 - x) * 0.5)))) * -2.0;
	double t_2 = t_1 * (t_1 - (0.5 * ((double) M_PI)));
	return fma(pow(((double) M_PI), 3.0), 0.125, pow(t_1, 3.0)) / ((pow(t_0, 3.0) + pow(t_2, 3.0)) / fma(t_0, t_0, ((t_2 * t_2) - (t_0 * t_2))));
}

Julia

function code(x)
	t_0 = Float64(0.25 * Float64(pi * pi))
	t_1 = Float64(Float64(Float64(0.5 * pi) - acos(sqrt(Float64(Float64(1.0 - x) * 0.5)))) * -2.0)
	t_2 = Float64(t_1 * Float64(t_1 - Float64(0.5 * pi)))
	return Float64(fma((pi ^ 3.0), 0.125, (t_1 ^ 3.0)) / Float64(Float64((t_0 ^ 3.0) + (t_2 ^ 3.0)) / fma(t_0, t_0, Float64(Float64(t_2 * t_2) - Float64(t_0 * t_2)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$0 * t$95$2), $MachinePrecision]), $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-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. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

   1. asin-acos-rev N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

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

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

   5. lift-PI.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

6. Applied rewrites 8.3%

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

8. Applied rewrites 8.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, 0.25, \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) - \left(0.5 \cdot \pi\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right)\right)}} \]

10. Applied rewrites 8.3%

    \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right)}^{3}\right)}{\frac{{\left(0.25 \cdot \left(\pi \cdot \pi\right)\right)}^{3} + {\left(\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2 - 0.5 \cdot \pi\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(0.25 \cdot \left(\pi \cdot \pi\right), 0.25 \cdot \left(\pi \cdot \pi\right), \left(\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2 - 0.5 \cdot \pi\right)\right) \cdot \left(\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2 - 0.5 \cdot \pi\right)\right) - \left(0.25 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2 - 0.5 \cdot \pi\right)\right)\right)}}} \]
11. Add Preprocessing

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\\ t_1 := t\_0 \cdot -2\\ \frac{\mathsf{fma}\left({t\_0}^{3}, -8, {\pi}^{3} \cdot 0.125\right)}{\mathsf{fma}\left(0.25, \pi \cdot \pi, t\_1 \cdot \left(t\_1 - 0.5 \cdot \pi\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* 0.5 PI) (acos (sqrt (* (- 1.0 x) 0.5)))))
        (t_1 (* t_0 -2.0)))
   (/
    (fma (pow t_0 3.0) -8.0 (* (pow PI 3.0) 0.125))
    (fma 0.25 (* PI PI) (* t_1 (- t_1 (* 0.5 PI)))))))

C

double code(double x) {
	double t_0 = (0.5 * ((double) M_PI)) - acos(sqrt(((1.0 - x) * 0.5)));
	double t_1 = t_0 * -2.0;
	return fma(pow(t_0, 3.0), -8.0, (pow(((double) M_PI), 3.0) * 0.125)) / fma(0.25, (((double) M_PI) * ((double) M_PI)), (t_1 * (t_1 - (0.5 * ((double) M_PI)))));
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 * pi) - acos(sqrt(Float64(Float64(1.0 - x) * 0.5))))
	t_1 = Float64(t_0 * -2.0)
	return Float64(fma((t_0 ^ 3.0), -8.0, Float64((pi ^ 3.0) * 0.125)) / fma(0.25, Float64(pi * pi), Float64(t_1 * Float64(t_1 - Float64(0.5 * pi)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * -2.0), $MachinePrecision]}, N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * -8.0 + N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(0.25 * N[(Pi * Pi), $MachinePrecision] + N[(t$95$1 * N[(t$95$1 - N[(0.5 * Pi), $MachinePrecision]), $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-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. asin-acos N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

   1. asin-acos-rev N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

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

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

   5. lift-PI.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

6. Applied rewrites 8.3%

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

8. Applied rewrites 8.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, 0.25, \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) - \left(0.5 \cdot \pi\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right)\right)}} \]

10. Applied rewrites 8.3%

    \[\leadsto \frac{\mathsf{fma}\left({\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right)}^{3}, -8, {\pi}^{3} \cdot 0.125\right)}{\color{blue}{\mathsf{fma}\left(0.25, \pi \cdot \pi, \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2\right) \cdot \left(\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \cdot -2 - 0.5 \cdot \pi\right)\right)}} \]
11. Add Preprocessing

Alternative 3: 8.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 8.3

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

3. Applied rewrites 8.3%

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

4. Taylor expanded in x around 0

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

   1. asin-acos-rev N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

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

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

   5. lift-PI.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

6. Applied rewrites 8.3%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.3

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

9. Applied rewrites 8.3%

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

Alternative 4: 6.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 6.8

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

4. Applied rewrites 6.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.8

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

7. Applied rewrites 6.8%

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

Alternative 5: 4.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 6.8

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

4. Applied rewrites 6.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 4.1%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

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 = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2025106 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))