bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.4%

Time: 21.5s

Alternatives: 10

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

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

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ \mathsf{fma}\left({\left(\sqrt[3]{t\_0}\right)}^{3}, e^{\mathsf{log1p}\left(t\_0\right)} + -1, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))))
   (fma
    (pow (cbrt t_0) 3.0)
    (+ (exp (log1p t_0)) -1.0)
    (- (expm1 (log1p (asin (- 1.0 x))))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	return fma(pow(cbrt(t_0), 3.0), (exp(log1p(t_0)) + -1.0), -expm1(log1p(asin((1.0 - x)))));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	return fma((cbrt(t_0) ^ 3.0), Float64(exp(log1p(t_0)) + -1.0), Float64(-expm1(log1p(asin(Float64(1.0 - x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + (-N[(Exp[N[Log[1 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 8.0%

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

   2. add-sqr-sqrt 6.2%

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

   3. fma-neg 6.2%

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

   4. div-inv 6.2%

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

   5. metadata-eval 6.2%

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

   6. div-inv 6.2%

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

   7. metadata-eval 6.2%

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

4. Applied egg-rr 6.2%

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

   1. expm1-log1p-u 6.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)}\right) \]

6. Applied egg-rr 6.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 11.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}} \cdot \sqrt[3]{\sqrt{\pi \cdot 0.5}}\right) \cdot \sqrt[3]{\sqrt{\pi \cdot 0.5}}}, \sqrt{\pi \cdot 0.5}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]

   2. pow3 11.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}}, \sqrt{\pi \cdot 0.5}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]

8. Applied egg-rr 11.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}}, \sqrt{\pi \cdot 0.5}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 11.0%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 0.5}\right)\right)}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]

   2. expm1-undefine 11.1%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}, \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 0.5}\right)} - 1}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]

10. Applied egg-rr 11.1%

    \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}, \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 0.5}\right)} - 1}, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]

11. Final simplification 11.1%

    \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\sqrt{\pi \cdot 0.5}}\right)}^{3}, e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 0.5}\right)} + -1, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) t_0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	return acos((1.0 - x)) + fma(-t_0, t_0, pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), t_0, (t_0 ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 8.0%

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

   2. *-un-lft-identity 8.0%

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

   3. add-sqr-sqrt 11.0%

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

   4. prod-diff 11.0%

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

   5. add-sqr-sqrt 11.1%

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

   6. fma-neg 11.1%

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

   7. *-un-lft-identity 11.1%

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

   8. acos-asin 11.1%

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

   9. add-sqr-sqrt 11.0%

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

4. Applied egg-rr 11.0%

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

   1. add-sqr-sqrt 11.0%

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

   2. pow2 11.0%

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

6. Applied egg-rr 11.1%

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

7. Final simplification 11.1%

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

Alternative 3: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float64(sqrt(pi) * sqrt(0.5)), sqrt(Float64(pi * 0.5)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 8.0%

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

   2. add-sqr-sqrt 6.2%

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

   3. fma-neg 6.2%

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

   4. div-inv 6.2%

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

   5. metadata-eval 6.2%

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

   6. div-inv 6.2%

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

   7. metadata-eval 6.2%

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

4. Applied egg-rr 6.2%

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

   1. sqrt-prod 11.1%

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

6. Applied egg-rr 11.1%

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

7. Final simplification 11.1%

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

Alternative 4: 9.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(e^{0.5 \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (- PI t_0) (* 2.0 (log (exp (* 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = 2.0 * log(exp((0.5 * t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = 2.0 * Math.log(Math.exp((0.5 * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = 2.0 * math.log(math.exp((0.5 * t_0)))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = Float64(2.0 * log(exp(Float64(0.5 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = pi - t_0;
	else
		tmp = 2.0 * log(exp((0.5 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], N[(2.0 * N[Log[N[Exp[N[(0.5 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 7.2%

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

      2. cancel-sign-sub-inv 7.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.4%

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

      5. sqr-neg 6.4%

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

      6. add-sqr-sqrt 6.4%

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

      7. add-sqr-sqrt 6.4%

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

   8. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.4%

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

      2. asin-acos 6.4%

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

      3. div-inv 6.4%

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

      4. metadata-eval 6.4%

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

      5. associate-+l- 6.4%

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

   10. Applied egg-rr 6.4%

       \[\leadsto \color{blue}{\pi \cdot 0.5 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)} \]
   11. Step-by-step derivation

       1. sub-neg 6.4%

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

       2. +-commutative 6.4%

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

       3. associate--r+ 6.4%

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

       4. sub-neg 6.4%

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

       5. remove-double-neg 6.4%

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

       6. distribute-lft-out 6.4%

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

       7. metadata-eval 6.4%

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

       8. *-rgt-identity 6.4%

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

   12. Simplified 6.4%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 56.4%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 56.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

      2. add-sqr-sqrt 56.3%

         \[\leadsto \log \color{blue}{\left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}} \cdot \sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

      3. log-prod 56.3%

         \[\leadsto \color{blue}{\log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right) + \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

   4. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right) + \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]
   5. Step-by-step derivation

      1. count-2 56.3%

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

   6. Simplified 56.3%

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

      1. pow1/2 56.3%

         \[\leadsto 2 \cdot \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.5}\right)} \]

      2. pow-exp 56.5%

         \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\cos^{-1} \left(1 - x\right) \cdot 0.5}\right)} \]

   8. Applied egg-rr 56.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\cos^{-1} \left(1 - x\right) \cdot 0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(e^{0.5 \cdot \cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 9.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\mathsf{log1p}\left(t\_0 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (- PI t_0) (pow E (log1p (+ t_0 -1.0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = pow(((double) M_E), log1p((t_0 + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = Math.pow(Math.E, Math.log1p((t_0 + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = math.pow(math.e, math.log1p((t_0 + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = exp(1) ^ log1p(Float64(t_0 + -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], N[Power[E, N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 7.2%

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

      2. cancel-sign-sub-inv 7.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.4%

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

      5. sqr-neg 6.4%

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

      6. add-sqr-sqrt 6.4%

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

      7. add-sqr-sqrt 6.4%

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

   8. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.4%

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

      2. asin-acos 6.4%

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

      3. div-inv 6.4%

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

      4. metadata-eval 6.4%

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

      5. associate-+l- 6.4%

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

   10. Applied egg-rr 6.4%

       \[\leadsto \color{blue}{\pi \cdot 0.5 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)} \]
   11. Step-by-step derivation

       1. sub-neg 6.4%

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

       2. +-commutative 6.4%

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

       3. associate--r+ 6.4%

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

       4. sub-neg 6.4%

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

       5. remove-double-neg 6.4%

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

       6. distribute-lft-out 6.4%

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

       7. metadata-eval 6.4%

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

       8. *-rgt-identity 6.4%

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

   12. Simplified 6.4%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 56.4%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 56.4%

         \[\leadsto \color{blue}{e^{\log \cos^{-1} \left(1 - x\right)}} \]

      2. *-un-lft-identity 56.4%

         \[\leadsto e^{\color{blue}{1 \cdot \log \cos^{-1} \left(1 - x\right)}} \]

      3. exp-prod 56.4%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\log \cos^{-1} \left(1 - x\right)}} \]

      4. exp-1-e 56.4%

         \[\leadsto {\color{blue}{e}}^{\log \cos^{-1} \left(1 - x\right)} \]

   4. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{{e}^{\log \cos^{-1} \left(1 - x\right)}} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 56.5%

         \[\leadsto {e}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \cos^{-1} \left(1 - x\right)\right)\right)\right)}} \]

      2. expm1-undefine 56.5%

         \[\leadsto {e}^{\left(\mathsf{log1p}\left(\color{blue}{e^{\log \cos^{-1} \left(1 - x\right)} - 1}\right)\right)} \]

      3. add-exp-log 56.5%

         \[\leadsto {e}^{\left(\mathsf{log1p}\left(\color{blue}{\cos^{-1} \left(1 - x\right)} - 1\right)\right)} \]

   6. Applied egg-rr 56.5%

      \[\leadsto {e}^{\color{blue}{\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right) - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

4. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. sub-neg 8.0%

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

6. Simplified 8.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 11.0%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 11.0%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

8. Applied egg-rr 11.0%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

9. Final simplification 11.0%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]
10. Add Preprocessing

Alternative 7: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

4. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
5. Step-by-step derivation

   1. sub-neg 8.0%

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

6. Simplified 8.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 11.0%

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

   2. pow2 11.0%

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

8. Applied egg-rr 11.0%

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

9. Final simplification 11.0%

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

Alternative 8: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= t_0 0.0) (- PI t_0) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = pi - t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 7.2%

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

      2. cancel-sign-sub-inv 7.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.4%

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

      5. sqr-neg 6.4%

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

      6. add-sqr-sqrt 6.4%

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

      7. add-sqr-sqrt 6.4%

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

   8. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.4%

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

      2. asin-acos 6.4%

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

      3. div-inv 6.4%

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

      4. metadata-eval 6.4%

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

      5. associate-+l- 6.4%

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

   10. Applied egg-rr 6.4%

       \[\leadsto \color{blue}{\pi \cdot 0.5 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)} \]
   11. Step-by-step derivation

       1. sub-neg 6.4%

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

       2. +-commutative 6.4%

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

       3. associate--r+ 6.4%

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

       4. sub-neg 6.4%

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

       5. remove-double-neg 6.4%

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

       6. distribute-lft-out 6.4%

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

       7. metadata-eval 6.4%

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

       8. *-rgt-identity 6.4%

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

   12. Simplified 6.4%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 56.4%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 6.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\pi \cdot 0.5 + \left(-1 - \sin^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (+ 1.0 (+ (* PI 0.5) (- -1.0 (asin (- 1.0 x)))))
   (- PI (acos (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + ((((double) M_PI) * 0.5) + (-1.0 - asin((1.0 - x))));
	} else {
		tmp = ((double) M_PI) - acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + ((Math.PI * 0.5) + (-1.0 - Math.asin((1.0 - x))));
	} else {
		tmp = Math.PI - Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 1.0 + ((math.pi * 0.5) + (-1.0 - math.asin((1.0 - x))))
	else:
		tmp = math.pi - math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(1.0 + Float64(Float64(pi * 0.5) + Float64(-1.0 - asin(Float64(1.0 - x)))));
	else
		tmp = Float64(pi - acos(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 1.0 + ((pi * 0.5) + (-1.0 - asin((1.0 - x))));
	else
		tmp = pi - acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(1.0 + N[(N[(Pi * 0.5), $MachinePrecision] + N[(-1.0 - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 8.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 8.0%

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

      2. sub-neg 8.0%

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

      3. div-inv 8.0%

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

      4. metadata-eval 8.0%

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

   4. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 8.0%

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

   6. Simplified 8.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 6.2%

         \[\leadsto \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)}\right) \]

   8. Applied egg-rr 8.0%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-undefine 8.0%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)} - 1\right)} \]

      2. associate--r- 8.0%

         \[\leadsto \color{blue}{\left(\pi \cdot 0.5 - e^{\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)}\right) + 1} \]

      3. log1p-undefine 8.0%

         \[\leadsto \left(\pi \cdot 0.5 - e^{\color{blue}{\log \left(1 + \sin^{-1} \left(1 - x\right)\right)}}\right) + 1 \]

      4. rem-exp-log 8.0%

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

   10. Applied egg-rr 8.0%

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

   if 1 < (-.f64 1 x)

   1. Initial program 8.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. acos-asin 8.0%

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

      2. sub-neg 8.0%

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

      3. div-inv 8.0%

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

      4. metadata-eval 8.0%

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

   4. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
   5. Step-by-step derivation

      1. sub-neg 8.0%

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

   6. Simplified 8.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 11.0%

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

      2. cancel-sign-sub-inv 11.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.9%

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

      5. sqr-neg 6.9%

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

      6. add-sqr-sqrt 6.9%

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

      7. add-sqr-sqrt 6.9%

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

   8. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.9%

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

      2. asin-acos 6.9%

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

      3. div-inv 6.9%

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

      4. metadata-eval 6.9%

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

      5. associate-+l- 6.9%

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

   10. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{\pi \cdot 0.5 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)} \]
   11. Step-by-step derivation

       1. sub-neg 6.9%

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

       2. +-commutative 6.9%

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

       3. associate--r+ 6.9%

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

       4. sub-neg 6.9%

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

       5. remove-double-neg 6.9%

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

       6. distribute-lft-out 6.9%

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

       7. metadata-eval 6.9%

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

       8. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

       \[\leadsto \color{blue}{\pi - \cos^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\pi \cdot 0.5 + \left(-1 - \sin^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 6.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Final simplification 8.0%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
4. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))