bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.3%

Time: 5.7s

Alternatives: 6

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: 6.7% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.0

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

5. Applied rewrites 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 7.0

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

7. Applied rewrites 7.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \color{blue}{\left(1 + -1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 5.2

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

10. Applied rewrites 5.2%

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

    1. lift-sqrt.f64 N/A

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

    2. pow1/2 N/A

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

    3. sqr-pow N/A

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

    4. lower-*.f64 N/A

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

    5. sqrt-pow1 N/A

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

    6. pow1/2 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lower-sqrt.f64 N/A

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

    9. sqrt-pow1 N/A

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

    10. pow1/2 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lower-sqrt.f64 10.7

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

12. Applied rewrites 10.7%

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

Alternative 2: 10.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 6.9

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

5. Applied rewrites 6.9%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 6.9

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

7. Applied rewrites 6.9%

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

8. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 6.7

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

10. Applied rewrites 6.7%

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

12. Applied rewrites 10.2%

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

13. Final simplification 10.2%

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

Developer Target 1: 100.0% accurate, 0.8× 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 2024226 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))