bug323 (missed optimization)

Percentage Accurate: 6.4% → 9.8%

Time: 24.4s

Alternatives: 11

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: 6.4% 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: 9.8% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (log
    (exp
     (+
      (acos (- 1.0 x))
      (fma (- (sqrt (pow (cbrt t_0) 3.0))) (sqrt t_0) t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return log(exp((acos((1.0 - x)) + fma(-sqrt(pow(cbrt(t_0), 3.0)), sqrt(t_0), t_0))));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return log(exp(Float64(acos(Float64(1.0 - x)) + fma(Float64(-sqrt((cbrt(t_0) ^ 3.0))), sqrt(t_0), t_0))))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. add-log-exp 7.2%

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

4. Applied egg-rr 7.2%

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

   1. acos-asin 7.3%

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

   2. *-un-lft-identity 7.3%

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

   3. add-sqr-sqrt 10.4%

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

   4. prod-diff 10.4%

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

   5. add-sqr-sqrt 7.2%

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

   6. fma-neg 7.2%

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

   7. *-un-lft-identity 7.2%

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

   8. acos-asin 7.2%

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

   9. add-sqr-sqrt 10.4%

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

6. Applied egg-rr 10.4%

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

   1. add-cube-cbrt 10.4%

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

   2. pow3 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

   \[\leadsto \log \left(e^{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)}\right) \]
10. Add Preprocessing

Alternative 2: 9.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (log (exp (+ (acos (- 1.0 x)) (fma (- t_1) t_1 t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return log(exp((acos((1.0 - x)) + fma(-t_1, t_1, t_0))));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return log(exp(Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0))))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. add-log-exp 7.2%

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

4. Applied egg-rr 7.2%

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

   1. acos-asin 7.3%

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

   2. *-un-lft-identity 7.3%

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

   3. add-sqr-sqrt 10.4%

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

   4. prod-diff 10.4%

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

   5. add-sqr-sqrt 7.2%

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

   6. fma-neg 7.2%

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

   7. *-un-lft-identity 7.2%

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

   8. acos-asin 7.2%

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

   9. add-sqr-sqrt 10.4%

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

6. Applied egg-rr 10.4%

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

7. Final simplification 10.4%

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

Alternative 3: 9.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. acos-asin 7.3%

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

   2. *-un-lft-identity 7.3%

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

   3. add-sqr-sqrt 10.4%

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

   4. prod-diff 10.4%

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

   5. add-sqr-sqrt 7.2%

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

   6. fma-neg 7.2%

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

   7. *-un-lft-identity 7.2%

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

   8. acos-asin 7.2%

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

   9. add-sqr-sqrt 10.4%

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

4. Applied egg-rr 10.4%

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

5. Final simplification 10.4%

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

Alternative 4: 9.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (pow (asin (- 1.0 x)) 1.5)) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(pow(asin((1.0 - x)), 1.5)), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.pow(Math.asin((1.0 - x)), 1.5)), 2.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt((asin(Float64(1.0 - x)) ^ 1.5)) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

4. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

6. Simplified 7.2%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

8. Applied egg-rr 10.4%

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

   1. add-cbrt-cube 10.4%

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

   2. pow1/3 10.4%

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

   3. add-sqr-sqrt 10.4%

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

   4. pow1 10.4%

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

   5. pow1/2 10.4%

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

   6. pow-prod-up 10.4%

      \[\leadsto \pi \cdot 0.5 - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \]

   7. metadata-eval 10.4%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

10. Applied egg-rr 10.4%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
11. Step-by-step derivation

    1. unpow1/3 10.4%

       \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]

12. Simplified 10.4%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]

13. Final simplification 10.4%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \]
14. Add Preprocessing

Alternative 5: 9.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17)
     (hypot (* PI 0.5) t_0)
     (- (* PI 0.5) (expm1 (log1p t_0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = hypot((((double) M_PI) * 0.5), t_0);
	} else {
		tmp = (((double) M_PI) * 0.5) - expm1(log1p(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.hypot((Math.PI * 0.5), t_0);
	} else {
		tmp = (Math.PI * 0.5) - Math.expm1(Math.log1p(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.hypot((math.pi * 0.5), t_0)
	else:
		tmp = (math.pi * 0.5) - math.expm1(math.log1p(t_0))
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = hypot(Float64(pi * 0.5), t_0);
	else
		tmp = Float64(Float64(pi * 0.5) - expm1(log1p(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. acos-asin 61.1%

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

      2. sub-neg 61.1%

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

      3. div-inv 61.1%

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

      4. metadata-eval 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. sub-neg 61.1%

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

   6. Simplified 61.1%

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

      1. expm1-log1p-u 61.1%

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

      2. expm1-undefine 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. expm1-define 61.1%

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

   10. Simplified 61.1%

       \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 9.8% 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 7.2%

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

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

4. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

6. Simplified 7.2%

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

   1. add-cube-cbrt 10.4%

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

   2. pow3 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

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

Alternative 7: 9.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

4. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

6. Simplified 7.2%

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

   1. add-sqr-sqrt 10.4%

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

   2. pow2 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

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

Alternative 8: 9.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (hypot (* PI 0.5) t_0) (- (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = hypot((((double) M_PI) * 0.5), t_0);
	} 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 (x <= 5.5e-17) {
		tmp = Math.hypot((Math.PI * 0.5), t_0);
	} else {
		tmp = (Math.PI * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.hypot((math.pi * 0.5), t_0)
	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 (x <= 5.5e-17)
		tmp = hypot(Float64(pi * 0.5), t_0);
	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 (x <= 5.5e-17)
		tmp = hypot((pi * 0.5), t_0);
	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[x, 5.5e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $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. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   6. Simplified 3.9%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 3.9%

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

      3. add-sqr-sqrt 3.9%

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

      4. cancel-sign-sub-inv 3.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.9%

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

      7. sqr-neg 3.9%

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

      8. add-sqr-sqrt 3.9%

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

      9. add-sqr-sqrt 3.9%

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

      10. difference-of-squares 3.9%

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

   8. Applied egg-rr 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. acos-asin 61.1%

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

      2. sub-neg 61.1%

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

      3. div-inv 61.1%

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

      4. metadata-eval 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. sub-neg 61.1%

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

   6. Simplified 61.1%

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

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 8.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (- PI (acos (- 1.0 x))) (- (* PI 0.5) (asin (- 1.0 x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $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. Step-by-step derivation

      1. add-log-exp 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. rem-log-exp 3.9%

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

      2. acos-asin 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. add-sqr-sqrt 7.3%

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

      6. cancel-sign-sub-inv 7.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.4%

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

      9. sqr-neg 6.4%

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

      10. add-sqr-sqrt 6.4%

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

      11. add-sqr-sqrt 6.4%

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

      12. asin-acos 6.4%

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

      13. div-inv 6.4%

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

      14. metadata-eval 6.4%

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

      15. associate-+r- 6.4%

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

   6. Applied egg-rr 6.4%

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

      1. fma-undefine 6.4%

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

      2. distribute-lft-out 6.4%

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

      3. metadata-eval 6.4%

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

      4. *-rgt-identity 6.4%

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

   8. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. acos-asin 61.1%

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

      2. sub-neg 61.1%

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

      3. div-inv 61.1%

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

      4. metadata-eval 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. sub-neg 61.1%

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

   6. Simplified 61.1%

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

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= x 5.5e-17) (- PI t_0) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI - t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi - t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi - t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi - t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi - t$95$0), $MachinePrecision], t$95$0]]

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. Step-by-step derivation

      1. add-log-exp 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. rem-log-exp 3.9%

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

      2. acos-asin 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

      5. add-sqr-sqrt 7.3%

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

      6. cancel-sign-sub-inv 7.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 6.4%

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

      9. sqr-neg 6.4%

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

      10. add-sqr-sqrt 6.4%

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

      11. add-sqr-sqrt 6.4%

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

      12. asin-acos 6.4%

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

      13. div-inv 6.4%

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

      14. metadata-eval 6.4%

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

      15. associate-+r- 6.4%

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

   6. Applied egg-rr 6.4%

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

      1. fma-undefine 6.4%

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

      2. distribute-lft-out 6.4%

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

      3. metadata-eval 6.4%

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

      4. *-rgt-identity 6.4%

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

   8. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

4. Final simplification 9.6%

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

Alternative 11: 6.4% 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 7.2%

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

3. Final simplification 7.2%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024053 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))