bug323 (missed optimization)

Percentage Accurate: 6.7% → 7.0%

Time: 4.8s

Alternatives: 4

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 7.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot \pi}{4} - {\left(0 - \sin^{-1} x\right)}^{2}}{\pi + \cos^{-1} x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (/ (* PI PI) 4.0) (pow (- 0.0 (asin x)) 2.0)) (+ PI (acos x))))

C

double code(double x) {
	return (((((double) M_PI) * ((double) M_PI)) / 4.0) - pow((0.0 - asin(x)), 2.0)) / (((double) M_PI) + acos(x));
}

Java

public static double code(double x) {
	return (((Math.PI * Math.PI) / 4.0) - Math.pow((0.0 - Math.asin(x)), 2.0)) / (Math.PI + Math.acos(x));
}

Python

def code(x):
	return (((math.pi * math.pi) / 4.0) - math.pow((0.0 - math.asin(x)), 2.0)) / (math.pi + math.acos(x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(pi * pi) / 4.0) - (Float64(0.0 - asin(x)) ^ 2.0)) / Float64(pi + acos(x)))
end

MATLAB

function tmp = code(x)
	tmp = (((pi * pi) / 4.0) - ((0.0 - asin(x)) ^ 2.0)) / (pi + acos(x));
end

Wolfram

code[x_] := N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] - N[Power[N[(0.0 - N[ArcSin[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Pi + N[ArcCos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.6%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.7%

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

5. Simplified 6.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
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 6.7%

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

7. Applied egg-rr 6.7%

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

9. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \pi}{4} - {\left(0 - \sin^{-1} x\right)}^{2}}{\cos^{-1} x + \pi}} \]

10. Final simplification 6.7%

    \[\leadsto \frac{\frac{\pi \cdot \pi}{4} - {\left(0 - \sin^{-1} x\right)}^{2}}{\pi + \cos^{-1} x} \]
11. Add Preprocessing

Alternative 2: 7.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\sin^{-1} x}^{2} - \frac{\pi \cdot \pi}{-4}}{\pi + \cos^{-1} x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (pow (asin x) 2.0) (/ (* PI PI) -4.0)) (+ PI (acos x))))

C

double code(double x) {
	return (pow(asin(x), 2.0) - ((((double) M_PI) * ((double) M_PI)) / -4.0)) / (((double) M_PI) + acos(x));
}

Java

public static double code(double x) {
	return (Math.pow(Math.asin(x), 2.0) - ((Math.PI * Math.PI) / -4.0)) / (Math.PI + Math.acos(x));
}

Python

def code(x):
	return (math.pow(math.asin(x), 2.0) - ((math.pi * math.pi) / -4.0)) / (math.pi + math.acos(x))

Julia

function code(x)
	return Float64(Float64((asin(x) ^ 2.0) - Float64(Float64(pi * pi) / -4.0)) / Float64(pi + acos(x)))
end

MATLAB

function tmp = code(x)
	tmp = ((asin(x) ^ 2.0) - ((pi * pi) / -4.0)) / (pi + acos(x));
end

Wolfram

code[x_] := N[(N[(N[Power[N[ArcSin[x], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(Pi * Pi), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / N[(Pi + N[ArcCos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.6%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.7%

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

5. Simplified 6.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
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 6.7%

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

7. Applied egg-rr 6.7%

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

9. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \pi}{4} - {\left(0 - \sin^{-1} x\right)}^{2}}{\cos^{-1} x + \pi}} \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \left(\mathsf{neg}\left({\left(0 - \sin^{-1} x\right)}^{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{acos.f64}\left(x\right)}, \mathsf{PI.f64}\left(\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left({\left(0 - \sin^{-1} x\right)}^{2}\right)\right) + \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{acos.f64}\left(x\right)}, \mathsf{PI.f64}\left(\right)\right)\right) \]

    3. unpow2 N/A

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

    4. fnmadd-define N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fnmadd}\left(\left(0 - \sin^{-1} x\right), \left(0 - \sin^{-1} x\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{acos.f64}\left(x\right)}, \mathsf{PI.f64}\left(\right)\right)\right) \]

11. Applied egg-rr 6.7%

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

12. Final simplification 6.7%

    \[\leadsto \frac{{\sin^{-1} x}^{2} - \frac{\pi \cdot \pi}{-4}}{\pi + \cos^{-1} x} \]
13. Add Preprocessing

Alternative 3: 6.9% 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 5.6%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.7%

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

5. Simplified 6.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(0 - x\right)} \]
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 6.7%

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

7. Applied egg-rr 6.7%

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

Alternative 4: 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 5.6%

   \[\cos^{-1} \left(1 - x\right) \]
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 3.9%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
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 2024148 
(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)))