bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 7.0s

Alternatives: 7

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}{2}, {\sin^{-1} \left(-1 + x\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right), t\_0, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (fma
     (pow (/ (PI) 2.0) 2.0)
     (/ (cbrt (pow (PI) 3.0)) 2.0)
     (pow (asin (+ -1.0 x)) 3.0))
    (fma (fma 0.5 (PI) t_0) t_0 (* (* (PI) (PI)) 0.25)))))

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

4. Applied rewrites 6.4%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right) + \frac{\mathsf{PI}\left(\right)}{2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. sqr-neg-rev N/A

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

   14. cube-unmult N/A

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

   15. cube-neg-rev N/A

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

   16. lift-pow.f64 N/A

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

6. Applied rewrites 10.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\mathsf{PI}\left(\right)}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\color{blue}{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 10.2%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\mathsf{PI}\left(\right)}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}} \]
10. Step-by-step derivation

    1. unpow1 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\color{blue}{{\mathsf{PI}\left(\right)}^{1}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{{\mathsf{PI}\left(\right)}^{\color{blue}{\left(3 \cdot \frac{1}{3}\right)}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

    3. pow-pow N/A

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\color{blue}{{\left({\mathsf{PI}\left(\right)}^{3}\right)}^{\frac{1}{3}}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

    4. pow1/3 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

    5. lower-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

    6. lower-pow.f64 10.2

       \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \]

11. Applied rewrites 10.2%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}}{2}, {\sin^{-1} \left(-\left(1 - x\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \]

12. Final simplification 10.2%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}, \frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}{2}, {\sin^{-1} \left(-1 + x\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right), \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \]
13. Add Preprocessing

Alternative 2: 10.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{-\mathsf{fma}\left(t\_0, t\_0, \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (- (fma t_0 t_0 (* (* -0.25 (PI)) (PI)))) (fma 0.5 (PI) t_0))))

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. flip-- N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

4. Applied rewrites 6.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\sin^{-1} \left(1 - x\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\sin^{-1} \left(1 - x\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)}} \]

7. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.25 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)}{-\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 10.2%

   \[\leadsto \frac{\mathsf{fma}\left(\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}{-\color{blue}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right)}} \]

10. Final simplification 10.2%

    \[\leadsto \frac{-\mathsf{fma}\left(\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(1 - x\right)\right)} \]
11. Add Preprocessing

Alternative 3: 10.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ \mathsf{fma}\left(t\_0, \frac{t\_0}{2}, \left(-t\_1\right) \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (sqrt (asin (- 1.0 x)))))
   (fma t_0 (/ t_0 2.0) (* (- t_1) t_1))))

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

4. Applied rewrites 6.4%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right) + \frac{\mathsf{PI}\left(\right)}{2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. sqr-neg-rev N/A

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

   14. cube-unmult N/A

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

   15. cube-neg-rev N/A

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

   16. lift-pow.f64 N/A

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

6. Applied rewrites 10.2%

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

8. Applied rewrites 10.2%

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

Alternative 4: 10.6% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 6.4

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

4. Applied rewrites 6.4%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 10.1

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

6. Applied rewrites 10.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]
7. Add Preprocessing

Alternative 5: 9.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(-x);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(Float64(-x));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(-x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 77.5%

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

Alternative 6: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

Alternative 7: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
6. 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 2024340 
(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)))