bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.5%

Time: 7.3s

Alternatives: 5

Speedup: 0.5×

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (fma t_0 (asin (- x 1.0)) (* (* (PI) (PI)) 0.25)) (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. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-PI.f64 6.4

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

4. Applied rewrites 6.4%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

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

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-asin.f64 N/A

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

   10. asin-neg-rev N/A

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

   11. lower-asin.f64 N/A

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

   12. lower-neg.f64 4.6

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

6. Applied rewrites 4.6%

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

7. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 10.0%

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

Alternative 2: 9.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999998:\\ \;\;\;\;e^{\log \cos^{-1} \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999999998)
   (exp (log (acos (- 1.0 x))))
   (acos (- x))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999998) {
		tmp = exp(log(acos((1.0 - x))));
	} else {
		tmp = acos(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - x) <= 0.9999999999999998d0) then
        tmp = exp(log(acos((1.0d0 - x))))
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999998) {
		tmp = Math.exp(Math.log(Math.acos((1.0 - x))));
	} else {
		tmp = Math.acos(-x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 0.9999999999999998:
		tmp = math.exp(math.log(math.acos((1.0 - x))))
	else:
		tmp = math.acos(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999998)
		tmp = exp(log(acos(Float64(1.0 - x))));
	else
		tmp = acos(Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 0.9999999999999998)
		tmp = exp(log(acos((1.0 - x))));
	else
		tmp = acos(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999998], N[Exp[N[Log[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcCos[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999978

   1. Initial program 58.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-asin.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. flip-- N/A

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

      10. lift-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-asin.f64 N/A

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

      13. acos-asin N/A

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

      14. lift-acos.f64 58.9

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

      15. unpow1 N/A

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

      16. pow-to-exp N/A

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

      17. lower-exp.f64 N/A

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

      18. rem-log-exp N/A

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

      19. pow-to-exp N/A

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

      20. unpow1 N/A

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

      21. lower-log.f64 58.9

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

   6. Applied rewrites 58.9%

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

   if 0.99999999999999978 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 3.9%

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

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

Alternative 3: 9.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = acos(-x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = acos(-x);
	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[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 58.9%

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

Alternative 4: 6.9% 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 5: 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 2024338 
(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)))