bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.5%

Time: 8.5s

Alternatives: 14

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: 7.1% 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: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \sqrt{\pi \cdot 0.5}\\ t\_0 + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt[3]{{\sin^{-1} \left(x \cdot \left(\frac{1}{x} + -1\right)\right)}^{1.5}}, \mathsf{fma}\left(t\_1, t\_1, -t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (sqrt (* PI 0.5))))
   (+
    t_0
    (fma
     (- (sqrt (asin (- 1.0 x))))
     (cbrt (pow (asin (* x (+ (/ 1.0 x) -1.0))) 1.5))
     (fma t_1 t_1 (- t_0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = sqrt((((double) M_PI) * 0.5));
	return t_0 + fma(-sqrt(asin((1.0 - x))), cbrt(pow(asin((x * ((1.0 / x) + -1.0))), 1.5)), fma(t_1, t_1, -t_0));
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = sqrt(Float64(pi * 0.5))
	return Float64(t_0 + fma(Float64(-sqrt(asin(Float64(1.0 - x)))), cbrt((asin(Float64(x * Float64(Float64(1.0 / x) + -1.0))) ^ 1.5)), fma(t_1, t_1, Float64(-t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 + N[((-N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) * N[Power[N[Power[N[ArcSin[N[(x * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision] + N[(t$95$1 * t$95$1 + (-t$95$0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

5. Taylor expanded in x around inf 10.3%

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

   1. asin-acos 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. add-sqr-sqrt 10.3%

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

   5. fma-neg 10.3%

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

7. Applied egg-rr 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. add-sqr-sqrt 5.7%

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

   3. pow1 5.7%

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

   4. pow1/2 5.7%

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

   5. pow-prod-up 10.3%

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

   6. sub-neg 10.3%

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

   7. metadata-eval 10.3%

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

   8. metadata-eval 10.3%

      \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt[3]{{\sin^{-1} \left(x \cdot \left(\frac{1}{x} + -1\right)\right)}^{\color{blue}{1.5}}}, \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]

9. Applied egg-rr 10.3%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \color{blue}{\sqrt[3]{{\sin^{-1} \left(x \cdot \left(\frac{1}{x} + -1\right)\right)}^{1.5}}}, \mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))) (t_1 (acos (- 1.0 x))))
   (+
    (+ -1.0 (+ 1.0 t_1))
    (fma
     (- (sqrt (asin (- 1.0 x))))
     (sqrt (asin (* x (+ (/ 1.0 x) -1.0))))
     (fma t_0 t_0 (- t_1))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = acos((1.0 - x));
	return (-1.0 + (1.0 + t_1)) + fma(-sqrt(asin((1.0 - x))), sqrt(asin((x * ((1.0 / x) + -1.0)))), fma(t_0, t_0, -t_1));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = acos(Float64(1.0 - x))
	return Float64(Float64(-1.0 + Float64(1.0 + t_1)) + fma(Float64(-sqrt(asin(Float64(1.0 - x)))), sqrt(asin(Float64(x * Float64(Float64(1.0 / x) + -1.0)))), fma(t_0, t_0, Float64(-t_1))))
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

5. Taylor expanded in x around inf 10.3%

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

   1. asin-acos 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. add-sqr-sqrt 10.3%

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

   5. fma-neg 10.3%

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

7. Applied egg-rr 10.3%

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

   1. expm1-log1p-u 10.3%

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

   2. expm1-undefine 10.3%

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

   3. log1p-undefine 10.3%

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

   4. rem-exp-log 10.3%

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

9. Applied egg-rr 10.3%

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

10. Final simplification 10.3%

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

Alternative 3: 10.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))) (t_1 (acos (- 1.0 x))))
   (+
    t_1
    (fma
     (- (sqrt (asin (- 1.0 x))))
     (sqrt (asin (* x (+ (/ 1.0 x) -1.0))))
     (fma t_0 t_0 (- t_1))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = acos((1.0 - x));
	return t_1 + fma(-sqrt(asin((1.0 - x))), sqrt(asin((x * ((1.0 / x) + -1.0)))), fma(t_0, t_0, -t_1));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = acos(Float64(1.0 - x))
	return Float64(t_1 + fma(Float64(-sqrt(asin(Float64(1.0 - x)))), sqrt(asin(Float64(x * Float64(Float64(1.0 / x) + -1.0)))), fma(t_0, t_0, Float64(-t_1))))
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

5. Taylor expanded in x around inf 10.3%

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

   1. asin-acos 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. add-sqr-sqrt 10.3%

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

   5. fma-neg 10.3%

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

7. Applied egg-rr 10.3%

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

8. Final simplification 10.3%

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

Alternative 4: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, \left(t\_0 - t\_0\right) - \sin^{-1} \left(1 - x\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (fma
    (pow (pow (* PI 0.5) 0.3333333333333333) 2.0)
    (cbrt (* PI 0.5))
    (- (- t_0 t_0) (asin (- 1.0 x))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	return fma(pow(pow((((double) M_PI) * 0.5), 0.3333333333333333), 2.0), cbrt((((double) M_PI) * 0.5)), ((t_0 - t_0) - asin((1.0 - x))));
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	return fma(((Float64(pi * 0.5) ^ 0.3333333333333333) ^ 2.0), cbrt(Float64(pi * 0.5)), Float64(Float64(t_0 - t_0) - asin(Float64(1.0 - x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(t$95$0 - t$95$0), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

   1. acos-asin 10.2%

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

   2. div-inv 10.2%

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

   3. metadata-eval 10.2%

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

   4. associate-+l- 10.2%

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

   5. add-cube-cbrt 5.1%

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

   6. fma-neg 5.1%

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

   7. pow2 5.1%

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

   8. fma-undefine 5.1%

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

6. Applied egg-rr 5.1%

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

   1. pow1/3 10.3%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\left(\sin^{-1} \left(1 - x\right) - \left(\cos^{-1} \left(1 - x\right) - \cos^{-1} \left(1 - x\right)\right)\right)\right) \]

8. Applied egg-rr 10.3%

   \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\left(\sin^{-1} \left(1 - x\right) - \left(\cos^{-1} \left(1 - x\right) - \cos^{-1} \left(1 - x\right)\right)\right)\right) \]

9. Final simplification 10.3%

   \[\leadsto \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, \left(\cos^{-1} \left(1 - x\right) - \cos^{-1} \left(1 - x\right)\right) - \sin^{-1} \left(1 - x\right)\right) \]
10. Add Preprocessing

Alternative 5: 10.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

5. Taylor expanded in x around inf 10.3%

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

6. Final simplification 10.3%

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

Alternative 6: 10.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

5. Taylor expanded in x around inf 10.3%

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

   1. asin-acos 10.3%

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

   2. div-inv 10.3%

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

   3. metadata-eval 10.3%

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

   4. add-sqr-sqrt 10.3%

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

   5. fma-neg 10.3%

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

7. Applied egg-rr 10.3%

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

8. Taylor expanded in x around 0 10.3%

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

9. Final simplification 10.3%

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

Alternative 7: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos x) (* 2.0 (log (exp (* (acos (- 1.0 x)) 0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(x);
	} else {
		tmp = 2.0 * log(exp((acos((1.0 - x)) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d-17) then
        tmp = acos(x)
    else
        tmp = 2.0d0 * log(exp((acos((1.0d0 - x)) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(x);
	} else {
		tmp = 2.0 * Math.log(Math.exp((Math.acos((1.0 - x)) * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(x)
	else:
		tmp = 2.0 * math.log(math.exp((math.acos((1.0 - x)) * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = Float64(2.0 * log(exp(Float64(acos(Float64(1.0 - x)) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = 2.0 * log(exp((acos((1.0 - x)) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[(2.0 * N[Log[N[Exp[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-sqr-sqrt 6.5%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.5%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 59.8%

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

      1. add-log-exp 59.8%

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

      2. add-sqr-sqrt 59.9%

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

      3. log-prod 59.8%

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

   4. Applied egg-rr 59.8%

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

      1. count-2 59.8%

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

   6. Simplified 59.8%

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

      1. pow1/2 59.8%

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

      2. pow-exp 59.8%

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

   8. Applied egg-rr 59.8%

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

Alternative 8: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos x) (log (exp (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(x);
	} else {
		tmp = log(exp(acos((1.0 - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d-17) then
        tmp = acos(x)
    else
        tmp = log(exp(acos((1.0d0 - x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(x);
	} else {
		tmp = Math.log(Math.exp(Math.acos((1.0 - x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(x)
	else:
		tmp = math.log(math.exp(math.acos((1.0 - x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = log(exp(acos(Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = log(exp(acos((1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[Log[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-sqr-sqrt 6.5%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.5%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 59.8%

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

      1. add-log-exp 59.8%

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

   4. Applied egg-rr 59.8%

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

Alternative 9: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. sub-neg 6.9%

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

   3. div-inv 6.9%

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

   4. metadata-eval 6.9%

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

4. Applied egg-rr 6.9%

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

   1. sub-neg 6.9%

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

6. Simplified 6.9%

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

   1. add-cbrt-cube 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

9. Final simplification 10.2%

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

Alternative 10: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. acos-asin 6.9%

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

   2. *-un-lft-identity 6.9%

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

   3. add-sqr-sqrt 10.2%

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

   4. prod-diff 10.2%

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

   5. add-sqr-sqrt 10.3%

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

   6. fma-neg 10.3%

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

   7. *-un-lft-identity 10.3%

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

   8. acos-asin 10.3%

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

   9. add-sqr-sqrt 10.2%

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

4. Applied egg-rr 10.2%

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

   1. acos-asin 10.2%

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

   2. div-inv 10.2%

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

   3. metadata-eval 10.2%

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

   4. associate-+l- 10.2%

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

   5. add-cube-cbrt 5.1%

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

   6. fma-neg 5.1%

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

   7. pow2 5.1%

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

   8. fma-undefine 5.1%

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

6. Applied egg-rr 5.1%

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

7. Taylor expanded in x around 0 10.2%

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

Alternative 11: 10.5% 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 6.9%

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

   1. acos-asin 6.9%

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

   2. sub-neg 6.9%

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

   3. div-inv 6.9%

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

   4. metadata-eval 6.9%

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

4. Applied egg-rr 6.9%

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

   1. sub-neg 6.9%

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

6. Simplified 6.9%

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

   1. add-cube-cbrt 10.2%

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

   2. pow3 10.2%

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

8. Applied egg-rr 10.2%

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

Alternative 12: 9.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.5e-17) (acos x) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(x);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d-17) then
        tmp = acos(x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.5%

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

      3. sqr-neg 6.5%

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

      4. sqrt-unprod 6.5%

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

      5. add-sqr-sqrt 6.5%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.5%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 59.8%

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

Alternative 13: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf 6.8%

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

   1. neg-mul-1 6.8%

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

5. Simplified 6.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.8%

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

   3. sqr-neg 6.8%

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

   4. sqrt-unprod 6.8%

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

   5. add-sqr-sqrt 6.8%

      \[\leadsto \cos^{-1} \color{blue}{x} \]

   6. *-un-lft-identity 6.8%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

7. Applied egg-rr 6.8%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
8. Step-by-step derivation

   1. *-lft-identity 6.8%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

9. Simplified 6.8%

   \[\leadsto \color{blue}{\cos^{-1} x} \]
10. Add Preprocessing

Alternative 14: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around 0 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Add Preprocessing

Developer Target 1: 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 2024130 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))