bug323 (missed optimization)

Percentage Accurate: 7.0% → 9.6%

Time: 5.5s

Alternatives: 5

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: 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: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi \cdot \pi}{4} - {\sin^{-1} x}^{2}}{\pi + \cos^{-1} x}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (/ (- (/ (* PI PI) 4.0) (pow (asin x) 2.0)) (+ PI (acos x)))
   (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((((double) M_PI) * ((double) M_PI)) / 4.0) - pow(asin(x), 2.0)) / (((double) M_PI) + acos(x));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((Math.PI * Math.PI) / 4.0) - Math.pow(Math.asin(x), 2.0)) / (Math.PI + Math.acos(x));
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = (((math.pi * math.pi) / 4.0) - math.pow(math.asin(x), 2.0)) / (math.pi + math.acos(x))
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(Float64(Float64(pi * pi) / 4.0) - (asin(x) ^ 2.0)) / Float64(pi + acos(x)));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (((pi * pi) / 4.0) - (asin(x) ^ 2.0)) / (pi + acos(x));
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] - N[Power[N[ArcSin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi + N[ArcCos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 6.6%

         \[\leadsto \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   5. Simplified 6.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. +-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\left(0 + x\right)\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left({0}^{3} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left(0 + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      6. +-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      7. cube-neg N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      8. sub0-neg N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      9. sqr-pow N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      10. unpow-prod-down N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(0 - x\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      11. sub0-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      12. sub0-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      13. sqr-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      14. pow-prod-down N/A

          \[\leadsto \cos^{-1} \left(\frac{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      15. sqr-pow N/A

          \[\leadsto \cos^{-1} \left(\frac{{x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      16. +-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\frac{0 + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      18. flip3-+ N/A

          \[\leadsto \cos^{-1} \left(0 + x\right) \]

      19. +-lft-identity N/A

          \[\leadsto \cos^{-1} x \]

      20. acos-lowering-acos.f64 6.6%

          \[\leadsto \mathsf{acos.f64}\left(x\right) \]

   7. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \pi}{4} - \left(0 - \sin^{-1} x\right) \cdot \left(0 - \sin^{-1} x\right)}{\cos^{-1} x + \pi}} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), 4\right), \left({\left(0 - \sin^{-1} x\right)}^{2}\right)\right), \mathsf{+.f64}\left(\mathsf{acos.f64}\left(x\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

       2. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), 4\right), \mathsf{pow.f64}\left(\left(0 - \sin^{-1} x\right), 2\right)\right), \mathsf{+.f64}\left(\mathsf{acos.f64}\left(x\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   11. Applied egg-rr 6.7%

       \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - \color{blue}{{\sin^{-1} x}^{2}}}{\cos^{-1} x + \pi} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 66.5%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi \cdot \pi}{4} - {\sin^{-1} x}^{2}}{\pi + \cos^{-1} x}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 7.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi + \cos^{-1} x\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_0}{\pi \cdot \pi}\right)}^{-2} - \frac{{\cos^{-1} x}^{4}}{{\left(\cos^{-1} x - \pi\right)}^{2}}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ PI (acos x))))
   (if (<= (- 1.0 x) 1.0)
     (acos (- 1.0 x))
     (/
      (-
       (pow (/ t_0 (* PI PI)) -2.0)
       (/ (pow (acos x) 4.0) (pow (- (acos x) PI) 2.0)))
      t_0))))

C

double code(double x) {
	double t_0 = ((double) M_PI) + acos(x);
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = acos((1.0 - x));
	} else {
		tmp = (pow((t_0 / (((double) M_PI) * ((double) M_PI))), -2.0) - (pow(acos(x), 4.0) / pow((acos(x) - ((double) M_PI)), 2.0))) / t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.PI + Math.acos(x);
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = (Math.pow((t_0 / (Math.PI * Math.PI)), -2.0) - (Math.pow(Math.acos(x), 4.0) / Math.pow((Math.acos(x) - Math.PI), 2.0))) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pi + math.acos(x)
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.acos((1.0 - x))
	else:
		tmp = (math.pow((t_0 / (math.pi * math.pi)), -2.0) - (math.pow(math.acos(x), 4.0) / math.pow((math.acos(x) - math.pi), 2.0))) / t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(pi + acos(x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = Float64(Float64((Float64(t_0 / Float64(pi * pi)) ^ -2.0) - Float64((acos(x) ^ 4.0) / (Float64(acos(x) - pi) ^ 2.0))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = pi + acos(x);
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = acos((1.0 - x));
	else
		tmp = (((t_0 / (pi * pi)) ^ -2.0) - ((acos(x) ^ 4.0) / ((acos(x) - pi) ^ 2.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi + N[ArcCos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(N[Power[N[(t$95$0 / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] - N[(N[Power[N[ArcCos[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[(N[ArcCos[x], $MachinePrecision] - Pi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.1%

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

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

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 6.9%

         \[\leadsto \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   5. Simplified 6.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. acos-neg N/A

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

      3. add-sqr-sqrt N/A

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

      4. fmm-def N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}, \left(\mathsf{neg}\left(\cos^{-1} x\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right), \left(\mathsf{neg}\left(\cos^{-1} x\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\mathsf{neg}\left(\cos^{-1} x\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(\cos^{-1} x\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(\cos^{-1} x\right)\right)\right) \]

      10. acos-asin N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} x\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\sin^{-1} x\right)\right)\right)\right)\right)\right) \]

      12. asin-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(0 - x\right)\right)\right)\right)\right) \]

      14. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(0 - x\right)\right)\right)\right) \]

      15. sub0-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      16. asin-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\sin^{-1} x\right)\right)\right)\right)\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} x\right)\right)\right) \]

      18. acos-asin N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\cos^{-1} x\right)\right) \]

      19. acos-lowering-acos.f64 6.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\mathsf{acos.f64}\left(x\right)\right)\right) \]

   7. Applied egg-rr 6.9%

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

   9. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \pi}{\cos^{-1} x + \pi} \cdot \frac{\pi \cdot \pi}{\cos^{-1} x + \pi} - \frac{{\cos^{-1} x}^{2}}{\cos^{-1} x - \pi} \cdot \frac{{\cos^{-1} x}^{2}}{\cos^{-1} x - \pi}}{\cos^{-1} x + \pi}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\cos^{-1} x + \mathsf{PI}\left(\right)} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\cos^{-1} x + \mathsf{PI}\left(\right)}\right), \left(\frac{{\cos^{-1} x}^{2}}{\cos^{-1} x - \mathsf{PI}\left(\right)} \cdot \frac{{\cos^{-1} x}^{2}}{\cos^{-1} x - \mathsf{PI}\left(\right)}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{acos.f64}\left(x\right)}, \mathsf{PI.f64}\left(\right)\right)\right) \]

   11. Applied egg-rr 6.9%

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

4. Final simplification 6.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\pi + \cos^{-1} x}{\pi \cdot \pi}\right)}^{-2} - \frac{{\cos^{-1} x}^{4}}{{\left(\cos^{-1} x - \pi\right)}^{2}}}{\pi + \cos^{-1} x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 9.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{acos.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 6.6%

         \[\leadsto \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   5. Simplified 6.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. +-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\left(0 + x\right)\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left({0}^{3} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left(0 + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      6. +-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      7. cube-neg N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      8. sub0-neg N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      9. sqr-pow N/A

         \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      10. unpow-prod-down N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(0 - x\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      11. sub0-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      12. sub0-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      13. sqr-neg N/A

          \[\leadsto \cos^{-1} \left(\frac{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      14. pow-prod-down N/A

          \[\leadsto \cos^{-1} \left(\frac{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      15. sqr-pow N/A

          \[\leadsto \cos^{-1} \left(\frac{{x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      16. +-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\frac{0 + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

      18. flip3-+ N/A

          \[\leadsto \cos^{-1} \left(0 + x\right) \]

      19. +-lft-identity N/A

          \[\leadsto \cos^{-1} x \]

      20. acos-lowering-acos.f64 6.6%

          \[\leadsto \mathsf{acos.f64}\left(x\right) \]

   7. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 66.5%

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

Alternative 4: 7.0% accurate, 1.0× speedup

Math

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

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(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(0 - x\right)\right) \]

   3. --lowering--.f64 6.9%

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

5. Simplified 6.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-lft-identity N/A

      \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\left(0 + x\right)\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \cos^{-1} \left(\mathsf{neg}\left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right)\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left({0}^{3} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left(\left(0 + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   6. +-lft-identity N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   7. cube-neg N/A

      \[\leadsto \cos^{-1} \left(\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   8. sub0-neg N/A

      \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   9. sqr-pow N/A

      \[\leadsto \cos^{-1} \left(\frac{{\left(0 - x\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   10. unpow-prod-down N/A

       \[\leadsto \cos^{-1} \left(\frac{{\left(\left(0 - x\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   11. sub0-neg N/A

       \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   12. sub0-neg N/A

       \[\leadsto \cos^{-1} \left(\frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   13. sqr-neg N/A

       \[\leadsto \cos^{-1} \left(\frac{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   14. pow-prod-down N/A

       \[\leadsto \cos^{-1} \left(\frac{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   15. sqr-pow N/A

       \[\leadsto \cos^{-1} \left(\frac{{x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   16. +-lft-identity N/A

       \[\leadsto \cos^{-1} \left(\frac{0 + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   17. metadata-eval N/A

       \[\leadsto \cos^{-1} \left(\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}\right) \]

   18. flip3-+ N/A

       \[\leadsto \cos^{-1} \left(0 + x\right) \]

   19. +-lft-identity N/A

       \[\leadsto \cos^{-1} x \]

   20. acos-lowering-acos.f64 6.9%

       \[\leadsto \mathsf{acos.f64}\left(x\right) \]

7. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{\cos^{-1} x} \]
8. 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 6.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 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 2024161 
(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)))