bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.6%

Time: 12.0s

Alternatives: 8

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: 7.0% 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: 10.6% accurate, 0.1× 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 6.0%

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

   1. acos-asin 6.0%

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

   2. flip-- 6.0%

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

   3. div-inv 6.0%

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

   4. metadata-eval 6.0%

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

   5. div-inv 6.0%

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

   6. metadata-eval 6.0%

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

   7. div-inv 6.0%

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

   8. metadata-eval 6.0%

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

3. Applied egg-rr 6.0%

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

   1. flip-- 6.0%

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

   2. add-sqr-sqrt 4.1%

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

   3. fma-neg 4.1%

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

5. Applied egg-rr 4.1%

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

   1. sqrt-prod 9.3%

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

7. Applied egg-rr 9.3%

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

8. Final simplification 9.3%

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

Alternative 2: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;{\left({t_0}^{0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ 1.0 (fabs (+ t_0 -1.0)))
     (pow (pow t_0 0.3333333333333333) 3.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + fabs((t_0 + -1.0));
	} else {
		tmp = pow(pow(t_0, 0.3333333333333333), 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + abs((t_0 + (-1.0d0)))
    else
        tmp = (t_0 ** 0.3333333333333333d0) ** 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + Math.abs((t_0 + -1.0));
	} else {
		tmp = Math.pow(Math.pow(t_0, 0.3333333333333333), 3.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + math.fabs((t_0 + -1.0))
	else:
		tmp = math.pow(math.pow(t_0, 0.3333333333333333), 3.0)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + abs(Float64(t_0 + -1.0)));
	else
		tmp = (t_0 ^ 0.3333333333333333) ^ 3.0;
	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 = 1.0 + abs((t_0 + -1.0));
	else
		tmp = (t_0 ^ 0.3333333333333333) ^ 3.0;
	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[(1.0 + N[Abs[N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[t$95$0, 0.3333333333333333], $MachinePrecision], 3.0], $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. flip-- 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. div-inv 3.9%

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

      6. metadata-eval 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 3.9%

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

      2. metadata-eval 3.9%

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

      3. div-inv 3.9%

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

      4. acos-asin 3.9%

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

      5. expm1-log1p-u 3.9%

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

      6. expm1-udef 3.9%

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

      7. log1p-udef 3.9%

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

      8. add-exp-log 3.9%

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

      9. associate--l+ 3.9%

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

      10. +-commutative 3.9%

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

      11. sub-neg 3.9%

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

      12. metadata-eval 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.4%

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

      3. pow2 6.4%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. rem-sqrt-square 6.4%

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

      3. +-commutative 6.4%

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

   9. Simplified 6.4%

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

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

   1. Initial program 63.1%

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

      1. add-cube-cbrt 62.9%

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

      2. pow3 62.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   3. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   4. Taylor expanded in x around 0 63.3%

      \[\leadsto {\color{blue}{\left({\cos^{-1} \left(1 - x\right)}^{0.3333333333333333}\right)}}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;{\left({\cos^{-1} \left(1 - x\right)}^{0.3333333333333333}\right)}^{3}\\ \end{array} \]

Alternative 3: 9.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (+ 1.0 (fabs (+ t_0 -1.0))) (log (exp t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + fabs((t_0 + -1.0));
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + abs((t_0 + (-1.0d0)))
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + math.fabs((t_0 + -1.0))
	else:
		tmp = math.log(math.exp(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(1.0 + abs(Float64(t_0 + -1.0)));
	else
		tmp = log(exp(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 = 1.0 + abs((t_0 + -1.0));
	else
		tmp = log(exp(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[(1.0 + N[Abs[N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. flip-- 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. div-inv 3.9%

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

      6. metadata-eval 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 3.9%

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

      2. metadata-eval 3.9%

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

      3. div-inv 3.9%

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

      4. acos-asin 3.9%

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

      5. expm1-log1p-u 3.9%

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

      6. expm1-udef 3.9%

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

      7. log1p-udef 3.9%

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

      8. add-exp-log 3.9%

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

      9. associate--l+ 3.9%

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

      10. +-commutative 3.9%

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

      11. sub-neg 3.9%

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

      12. metadata-eval 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.4%

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

      3. pow2 6.4%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. rem-sqrt-square 6.4%

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

      3. +-commutative 6.4%

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

   9. Simplified 6.4%

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

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

   1. Initial program 63.1%

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

      1. add-log-exp 63.1%

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

   3. Applied egg-rr 63.1%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \]

Alternative 4: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. acos-asin 6.0%

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

   2. flip-- 6.0%

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

   3. div-inv 6.0%

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

   4. metadata-eval 6.0%

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

   5. div-inv 6.0%

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

   6. metadata-eval 6.0%

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

   7. div-inv 6.0%

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

   8. metadata-eval 6.0%

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

3. Applied egg-rr 6.0%

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

   1. flip-- 6.0%

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

   2. add-sqr-sqrt 4.1%

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

   3. fma-neg 4.1%

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

5. Applied egg-rr 4.1%

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

6. Taylor expanded in x around 0 9.3%

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

7. Final simplification 9.3%

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

Alternative 5: 7.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= (- 1.0 x) 1.0) (+ 1.0 (pow (cbrt t_0) 3.0)) (+ 1.0 (fabs t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + pow(cbrt(t_0), 3.0);
	} else {
		tmp = 1.0 + fabs(t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + Math.pow(Math.cbrt(t_0), 3.0);
	} else {
		tmp = 1.0 + Math.abs(t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(1.0 + (cbrt(t_0) ^ 3.0));
	else
		tmp = Float64(1.0 + abs(t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(1.0 + N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.0%

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

      1. acos-asin 6.0%

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

      2. flip-- 6.0%

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

      3. div-inv 6.0%

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

      4. metadata-eval 6.0%

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

      5. div-inv 6.0%

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

      6. metadata-eval 6.0%

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

      7. div-inv 6.0%

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

      8. metadata-eval 6.0%

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

   3. Applied egg-rr 6.0%

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

      1. flip-- 6.0%

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

      2. metadata-eval 6.0%

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

      3. div-inv 6.0%

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

      4. acos-asin 6.0%

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

      5. expm1-log1p-u 6.0%

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

      6. expm1-udef 6.0%

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

      7. log1p-udef 6.0%

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

      8. add-exp-log 6.0%

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

      9. associate--l+ 6.0%

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

      10. +-commutative 6.0%

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

      11. sub-neg 6.0%

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

      12. metadata-eval 6.0%

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

   5. Applied egg-rr 6.0%

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

      1. add-cube-cbrt 6.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right) + -1} \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}\right) \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}} + 1 \]

      2. pow3 6.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}} + 1 \]

   7. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}} + 1 \]

   if 1 < (-.f64 1 x)

   1. Initial program 6.0%

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

      1. acos-asin 6.0%

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

      2. flip-- 6.0%

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

      3. div-inv 6.0%

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

      4. metadata-eval 6.0%

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

      5. div-inv 6.0%

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

      6. metadata-eval 6.0%

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

      7. div-inv 6.0%

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

      8. metadata-eval 6.0%

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

   3. Applied egg-rr 6.0%

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

      1. flip-- 6.0%

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

      2. metadata-eval 6.0%

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

      3. div-inv 6.0%

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

      4. acos-asin 6.0%

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

      5. expm1-log1p-u 6.0%

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

      6. expm1-udef 6.0%

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

      7. log1p-udef 6.0%

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

      8. add-exp-log 6.0%

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

      9. associate--l+ 6.0%

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

      10. +-commutative 6.0%

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

      11. sub-neg 6.0%

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

      12. metadata-eval 6.0%

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

   5. Applied egg-rr 6.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. pow2 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. unpow2 6.7%

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

      2. rem-sqrt-square 6.7%

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

      3. +-commutative 6.7%

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

   9. Simplified 6.7%

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

4. Final simplification 6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + {\left(\sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]

Alternative 6: 9.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + abs((t_0 + (-1.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + Math.abs((t_0 + -1.0));
	} 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 = 1.0 + math.fabs((t_0 + -1.0))
	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(1.0 + abs(Float64(t_0 + -1.0)));
	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 = 1.0 + abs((t_0 + -1.0));
	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[(1.0 + N[Abs[N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]), $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. flip-- 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. div-inv 3.9%

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

      6. metadata-eval 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 3.9%

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

      2. metadata-eval 3.9%

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

      3. div-inv 3.9%

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

      4. acos-asin 3.9%

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

      5. expm1-log1p-u 3.9%

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

      6. expm1-udef 3.9%

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

      7. log1p-udef 3.9%

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

      8. add-exp-log 3.9%

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

      9. associate--l+ 3.9%

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

      10. +-commutative 3.9%

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

      11. sub-neg 3.9%

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

      12. metadata-eval 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.4%

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

      3. pow2 6.4%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. rem-sqrt-square 6.4%

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

      3. +-commutative 6.4%

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

   9. Simplified 6.4%

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

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

   1. Initial program 63.1%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 7: 9.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (+ (* PI 0.5) (asin (- 1.0 x))) 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) * 0.5) + asin((1.0 - x));
	} 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 * 0.5) + Math.asin((1.0 - x));
	} 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 * 0.5) + math.asin((1.0 - x))
	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(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	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 * 0.5) + asin((1.0 - x));
	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[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. flip-- 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. div-inv 3.9%

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

      6. metadata-eval 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 3.9%

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

      2. add-cube-cbrt 2.0%

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

      3. fma-neg 2.0%

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

      4. pow2 2.0%

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

   5. Applied egg-rr 2.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)} \]
   6. Step-by-step derivation

      1. fma-udef 2.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{2} \cdot \sqrt[3]{\pi \cdot 0.5} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      2. unpow2 2.0%

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

      3. add-cube-cbrt 3.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 6.4%

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

      6. sqr-neg 6.4%

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

      7. sqrt-prod 6.4%

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

      8. add-sqr-sqrt 6.4%

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

   7. Applied egg-rr 6.4%

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

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

   1. Initial program 63.1%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 8: 7.0% 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 6.0%

   \[\cos^{-1} \left(1 - x\right) \]

2. Final simplification 6.0%

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

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 2023275 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))