bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 21.8s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. *-un-lft-identity 7.1%

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

   3. add-sqr-sqrt 10.7%

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

   4. prod-diff 10.7%

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

   5. add-sqr-sqrt 10.8%

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

   6. fma-neg 10.8%

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

   7. *-un-lft-identity 10.8%

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

   8. acos-asin 10.8%

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

   9. add-sqr-sqrt 10.7%

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

3. Applied egg-rr 10.7%

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

   1. add-cube-cbrt 5.3%

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

   2. unpow2 5.3%

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

   3. *-commutative 5.3%

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

   4. add-sqr-sqrt 5.3%

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

   5. pow2 5.3%

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

   6. *-commutative 5.3%

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

   7. unpow2 5.3%

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

   8. add-cube-cbrt 10.8%

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

5. Applied egg-rr 10.8%

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

6. Final simplification 10.8%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. add-cube-cbrt 5.3%

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

   3. fma-neg 5.3%

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

   4. pow2 5.3%

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

   5. div-inv 5.3%

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

   6. metadata-eval 5.3%

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

   7. div-inv 5.3%

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

   8. metadata-eval 5.3%

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

3. Applied egg-rr 5.3%

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

   1. pow1/3 10.8%

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

5. Applied egg-rr 10.8%

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

6. Final simplification 10.8%

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

Alternative 3: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. add-cube-cbrt 10.8%

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

   2. pow3 10.8%

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

7. Applied egg-rr 10.8%

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

8. Final simplification 10.8%

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

Alternative 4: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;t_0 + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(0.3333333333333333 \cdot \left(\pi \cdot 0.5 - t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= (acos (- 1.0 x)) 0.0)
     (+ t_0 (* PI 0.5))
     (* 3.0 (* 0.3333333333333333 (- (* PI 0.5) t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (acos((1.0 - x)) <= 0.0) {
		tmp = t_0 + (((double) M_PI) * 0.5);
	} else {
		tmp = 3.0 * (0.3333333333333333 * ((((double) M_PI) * 0.5) - t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (Math.acos((1.0 - x)) <= 0.0) {
		tmp = t_0 + (Math.PI * 0.5);
	} else {
		tmp = 3.0 * (0.3333333333333333 * ((Math.PI * 0.5) - t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if math.acos((1.0 - x)) <= 0.0:
		tmp = t_0 + (math.pi * 0.5)
	else:
		tmp = 3.0 * (0.3333333333333333 * ((math.pi * 0.5) - t_0))
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (acos(Float64(1.0 - x)) <= 0.0)
		tmp = Float64(t_0 + Float64(pi * 0.5));
	else
		tmp = Float64(3.0 * Float64(0.3333333333333333 * Float64(Float64(pi * 0.5) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if (acos((1.0 - x)) <= 0.0)
		tmp = t_0 + (pi * 0.5);
	else
		tmp = 3.0 * (0.3333333333333333 * ((pi * 0.5) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.0], N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(0.3333333333333333 * N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   5. Simplified 3.8%

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

      1. sub-neg 3.8%

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

      2. add-cube-cbrt 7.7%

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

      3. unpow2 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

      8. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 6.7%

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

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

   1. Initial program 63.1%

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

      1. acos-asin 63.3%

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

      2. sub-neg 63.3%

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

      3. div-inv 63.3%

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

      4. metadata-eval 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. sub-neg 63.3%

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

   5. Simplified 63.3%

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

      1. metadata-eval 63.3%

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

      2. div-inv 63.3%

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

      3. acos-asin 63.1%

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

      4. add-log-exp 63.2%

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

      5. add-cube-cbrt 63.0%

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

      6. pow3 63.0%

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

      7. exp-to-pow 63.0%

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

      8. add-log-exp 63.0%

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

      9. pow1/3 62.9%

         \[\leadsto \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)} \cdot 3 \]

      10. log-pow 63.2%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)} \cdot 3 \]

      11. add-log-exp 63.2%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right) \cdot 3 \]

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) \cdot 3} \]
   8. Step-by-step derivation

      1. acos-asin 63.3%

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

      2. sub-neg 63.3%

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

      3. div-inv 63.3%

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

      4. metadata-eval 63.3%

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

   9. Applied egg-rr 63.3%

      \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left(\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)\right)}\right) \cdot 3 \]
   10. Step-by-step derivation

       1. sub-neg 63.3%

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

   11. Simplified 63.3%

       \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\right)}\right) \cdot 3 \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(0.3333333333333333 \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\right)\right)\\ \end{array} \]

Alternative 5: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 3 \cdot \left(t_0 \cdot 0.3333333333333333\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (asin (- 1.0 x)) (* PI 0.5))
     (+ (+ 1.0 (* 3.0 (* t_0 0.3333333333333333))) -1.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = asin((1.0 - x)) + (((double) M_PI) * 0.5);
	} else {
		tmp = (1.0 + (3.0 * (t_0 * 0.3333333333333333))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.asin((1.0 - x)) + (Math.PI * 0.5);
	} else {
		tmp = (1.0 + (3.0 * (t_0 * 0.3333333333333333))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.asin((1.0 - x)) + (math.pi * 0.5)
	else:
		tmp = (1.0 + (3.0 * (t_0 * 0.3333333333333333))) + -1.0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(asin(Float64(1.0 - x)) + Float64(pi * 0.5));
	else
		tmp = Float64(Float64(1.0 + Float64(3.0 * Float64(t_0 * 0.3333333333333333))) + -1.0);
	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 = asin((1.0 - x)) + (pi * 0.5);
	else
		tmp = (1.0 + (3.0 * (t_0 * 0.3333333333333333))) + -1.0;
	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[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(3.0 * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   5. Simplified 3.8%

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

      1. sub-neg 3.8%

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

      2. add-cube-cbrt 7.7%

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

      3. unpow2 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

      8. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 6.7%

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

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

   1. Initial program 63.1%

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

      1. expm1-log1p-u 63.2%

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

      2. expm1-udef 63.1%

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

      3. log1p-udef 63.0%

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

      4. rem-exp-log 63.0%

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

   3. Applied egg-rr 63.0%

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

      1. add-log-exp 63.3%

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

      2. add-cube-cbrt 63.0%

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

      3. log-prod 63.0%

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

      4. pow2 63.0%

         \[\leadsto \left(1 + \left(\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)\right) - 1 \]

   5. Applied egg-rr 63.0%

      \[\leadsto \left(1 + \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)}\right) - 1 \]
   6. Step-by-step derivation

      1. log-pow 63.0%

         \[\leadsto \left(1 + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)\right) - 1 \]

      2. distribute-lft1-in 63.0%

         \[\leadsto \left(1 + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}\right) - 1 \]

      3. metadata-eval 63.0%

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

      4. *-commutative 63.0%

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

   7. Simplified 63.0%

      \[\leadsto \left(1 + \color{blue}{\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \cdot 3}\right) - 1 \]
   8. Step-by-step derivation

      1. pow1/3 62.9%

         \[\leadsto \left(1 + \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)} \cdot 3\right) - 1 \]

      2. log-pow 63.3%

         \[\leadsto \left(1 + \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)} \cdot 3\right) - 1 \]

      3. add-log-exp 63.3%

         \[\leadsto \left(1 + \left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right) \cdot 3\right) - 1 \]

   9. Applied egg-rr 63.3%

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

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right)\right) + -1\\ \end{array} \]

Alternative 6: 9.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= (acos (- 1.0 x)) 0.0) (+ t_0 (* PI 0.5)) (- (* PI 0.5) t_0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   5. Simplified 3.8%

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

      1. sub-neg 3.8%

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

      2. add-cube-cbrt 7.7%

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

      3. unpow2 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

      8. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 6.7%

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

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

   1. Initial program 63.1%

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

      1. acos-asin 63.3%

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

      2. sub-neg 63.3%

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

      3. div-inv 63.3%

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

      4. metadata-eval 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. sub-neg 63.3%

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

   5. Simplified 63.3%

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

4. Final simplification 9.8%

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

Alternative 7: 6.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + 3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right)\right) + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ 1.0 (* 3.0 (* (acos (- 1.0 x)) 0.3333333333333333))) -1.0))

C

double code(double x) {
	return (1.0 + (3.0 * (acos((1.0 - x)) * 0.3333333333333333))) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + (3.0d0 * (acos((1.0d0 - x)) * 0.3333333333333333d0))) + (-1.0d0)
end function

Java

public static double code(double x) {
	return (1.0 + (3.0 * (Math.acos((1.0 - x)) * 0.3333333333333333))) + -1.0;
}

Python

def code(x):
	return (1.0 + (3.0 * (math.acos((1.0 - x)) * 0.3333333333333333))) + -1.0

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(3.0 * Float64(acos(Float64(1.0 - x)) * 0.3333333333333333))) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (3.0 * (acos((1.0 - x)) * 0.3333333333333333))) + -1.0;
end

Wolfram

code[x_] := N[(N[(1.0 + N[(3.0 * N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. expm1-log1p-u 7.1%

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

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. rem-exp-log 7.1%

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

3. Applied egg-rr 7.1%

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

   1. add-log-exp 7.1%

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

   2. add-cube-cbrt 7.1%

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

   3. log-prod 7.1%

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

   4. pow2 7.1%

      \[\leadsto \left(1 + \left(\log \color{blue}{\left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)\right) - 1 \]

5. Applied egg-rr 7.1%

   \[\leadsto \left(1 + \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)}\right) - 1 \]
6. Step-by-step derivation

   1. log-pow 7.1%

      \[\leadsto \left(1 + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)\right) - 1 \]

   2. distribute-lft1-in 7.1%

      \[\leadsto \left(1 + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}\right) - 1 \]

   3. metadata-eval 7.1%

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

   4. *-commutative 7.1%

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

7. Simplified 7.1%

   \[\leadsto \left(1 + \color{blue}{\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \cdot 3}\right) - 1 \]
8. Step-by-step derivation

   1. pow1/3 7.1%

      \[\leadsto \left(1 + \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)} \cdot 3\right) - 1 \]

   2. log-pow 7.1%

      \[\leadsto \left(1 + \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)} \cdot 3\right) - 1 \]

   3. add-log-exp 7.1%

      \[\leadsto \left(1 + \left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right) \cdot 3\right) - 1 \]

9. Applied egg-rr 7.1%

   \[\leadsto \left(1 + \color{blue}{\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)} \cdot 3\right) - 1 \]

10. Final simplification 7.1%

    \[\leadsto \left(1 + 3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right)\right) + -1 \]

Alternative 8: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 3.0 (* (acos (- 1.0 x)) 0.3333333333333333)))

C

double code(double x) {
	return 3.0 * (acos((1.0 - x)) * 0.3333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * (acos((1.0d0 - x)) * 0.3333333333333333d0)
end function

Java

public static double code(double x) {
	return 3.0 * (Math.acos((1.0 - x)) * 0.3333333333333333);
}

Python

def code(x):
	return 3.0 * (math.acos((1.0 - x)) * 0.3333333333333333)

Julia

function code(x)
	return Float64(3.0 * Float64(acos(Float64(1.0 - x)) * 0.3333333333333333))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * (acos((1.0 - x)) * 0.3333333333333333);
end

Wolfram

code[x_] := N[(3.0 * N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. acos-asin 7.1%

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

   2. sub-neg 7.1%

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

   3. div-inv 7.1%

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

   4. metadata-eval 7.1%

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

3. Applied egg-rr 7.1%

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

   1. sub-neg 7.1%

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

5. Simplified 7.1%

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

   1. metadata-eval 7.1%

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

   2. div-inv 7.1%

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

   3. acos-asin 7.1%

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

   4. add-log-exp 7.1%

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

   5. add-cube-cbrt 7.1%

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

   6. pow3 7.1%

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

   7. exp-to-pow 7.1%

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

   8. add-log-exp 7.1%

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

   9. pow1/3 7.1%

      \[\leadsto \log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)} \cdot 3 \]

   10. log-pow 7.1%

       \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)} \cdot 3 \]

   11. add-log-exp 7.1%

       \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right) \cdot 3 \]

7. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) \cdot 3} \]

8. Final simplification 7.1%

   \[\leadsto 3 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right) \]

Alternative 9: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Final simplification 7.1%

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

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 2023310 
(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)))