cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 99.3%

Time: 11.0s

Alternatives: 7

Speedup: 17.1×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(x \cdot 0.5\right)}{x \cdot \frac{x}{\sin x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (tan (* x 0.5)) (* x (/ x (sin x)))))

C

double code(double x) {
	return tan((x * 0.5)) / (x * (x / sin(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) / (x * (x / sin(x)))
end function

Java

public static double code(double x) {
	return Math.tan((x * 0.5)) / (x * (x / Math.sin(x)));
}

Python

def code(x):
	return math.tan((x * 0.5)) / (x * (x / math.sin(x)))

Julia

function code(x)
	return Float64(tan(Float64(x * 0.5)) / Float64(x * Float64(x / sin(x))))
end

MATLAB

function tmp = code(x)
	tmp = tan((x * 0.5)) / (x * (x / sin(x)));
end

Wolfram

code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}{\color{blue}{x \cdot x}} \]

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

   11. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin x} \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \sin x \cdot \color{blue}{\frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

   13. lower-sin.f64 N/A

       \[\leadsto \sin x \cdot \frac{\color{blue}{\sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \cos x\right)} \]

   15. associate-*l* N/A

       \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

   16. lower-*.f64 N/A

       \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \sin x \cdot \frac{\sin x}{x \cdot \color{blue}{\left(x \cdot \left(1 + \cos x\right)\right)}} \]

   18. lower-+.f64 75.4

       \[\leadsto \sin x \cdot \frac{\sin x}{x \cdot \left(x \cdot \color{blue}{\left(1 + \cos x\right)}\right)} \]

4. Applied egg-rr 75.4%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

5. Taylor expanded in x around inf

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

   1. associate-/l/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. hang-0p-tan N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{{x}^{2}} \]

   8. lower-tan.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{{x}^{2}} \]

   9. *-rgt-identity N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

       \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x \cdot x}} \]

   14. lower-*.f64 75.3

       \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{\color{blue}{x \cdot x}} \]

7. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x \cdot x}} \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}{x \cdot x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\sin x \cdot \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

   3. lift-tan.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(x \cdot \frac{1}{2}\right)}{x}} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}{x}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

   10. lower-/.f64 99.6

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin x}{x}} \cdot \tan \left(x \cdot 0.5\right)}} \]

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot 0.5\right)}}} \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\frac{\color{blue}{\sin x}}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin x}{x}} \cdot \tan \left(x \cdot \frac{1}{2}\right)}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}} \]

    4. lift-tan.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \color{blue}{\tan \left(x \cdot \frac{1}{2}\right)}}} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

    6. lift-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

    7. /-rgt-identity N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}{1}}} \]

    8. clear-num N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

    9. lift-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin x}{x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

    11. associate-/r* N/A

        \[\leadsto \frac{1}{\color{blue}{\frac{\frac{x}{\frac{\sin x}{x}}}{\tan \left(x \cdot \frac{1}{2}\right)}}} \]

    12. clear-num N/A

        \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{\frac{x}{\frac{\sin x}{x}}}} \]

    13. lower-/.f64 N/A

        \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{\frac{x}{\frac{\sin x}{x}}}} \]

    14. div-inv N/A

        \[\leadsto \frac{\tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x \cdot \frac{1}{\frac{\sin x}{x}}}} \]

    15. lift-/.f64 N/A

        \[\leadsto \frac{\tan \left(x \cdot \frac{1}{2}\right)}{x \cdot \frac{1}{\color{blue}{\frac{\sin x}{x}}}} \]

    16. clear-num N/A

        \[\leadsto \frac{\tan \left(x \cdot \frac{1}{2}\right)}{x \cdot \color{blue}{\frac{x}{\sin x}}} \]

    17. lower-*.f64 N/A

        \[\leadsto \frac{\tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

    18. lower-/.f64 99.7

        \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{x \cdot \color{blue}{\frac{x}{\sin x}}} \]

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x \cdot \frac{x}{\sin x}}} \]
12. Add Preprocessing

Alternative 2: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-5)
   (fma x (* x -0.041666666666666664) 0.5)
   (* (/ 1.0 (* x x)) (* (tan (* x 0.5)) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-5) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 / (x * x)) * (tan((x * 0.5)) * sin(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-5)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(x * x)) * Float64(tan(Float64(x * 0.5)) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e-5], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000033e-5

   1. Initial program 31.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 70.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 4.00000000000000033e-5 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}{\color{blue}{x \cdot x}} \]

      4. associate-/l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. 1-sub-cos N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

      11. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \sin x \cdot \color{blue}{\frac{\sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sin x \cdot \frac{\color{blue}{\sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \cos x\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \frac{\sin x}{\color{blue}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

      17. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \frac{\sin x}{x \cdot \color{blue}{\left(x \cdot \left(1 + \cos x\right)\right)}} \]

      18. lower-+.f64 98.7

          \[\leadsto \sin x \cdot \frac{\sin x}{x \cdot \left(x \cdot \color{blue}{\left(1 + \cos x\right)}\right)} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{x \cdot \left(x \cdot \left(1 + \cos x\right)\right)}} \]

   5. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. hang-0p-tan N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{{x}^{2}} \]

      8. lower-tan.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{{x}^{2}} \]

      9. *-rgt-identity N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x \cdot x}} \]

      14. lower-*.f64 99.6

          \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{\color{blue}{x \cdot x}} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x \cdot x}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x} \cdot \tan \left(x \cdot \frac{1}{2}\right)}{x \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

      3. lift-tan.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x \cdot x}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}}} \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot \left(\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot \left(\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)\right)} \]

      9. lower-/.f64 99.8

         \[\leadsto \color{blue}{\frac{1}{x \cdot x}} \cdot \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0051:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0051)
   (fma x (* x -0.041666666666666664) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0051) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0051)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0051], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051000000000000004

   1. Initial program 31.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 70.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 0.0051000000000000004 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.7%

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

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos x} + -1}{x} \cdot \frac{-1}{x} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos x + -1}}{x} \cdot \frac{-1}{x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos x + -1}{x}} \cdot \frac{-1}{x} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos x + -1}{x} \cdot -1}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos x + -1}{x} \cdot -1}{x}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos x + -1}{x}} \cdot -1}{x} \]

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 99.7

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

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0051:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0051)
   (fma x (* x -0.041666666666666664) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0051) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0051)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0051], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051000000000000004

   1. Initial program 31.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 70.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 0.0051000000000000004 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (fma x (* x -0.16666666666666666) -2.0)))

C

double code(double x) {
	return -1.0 / fma(x, (x * -0.16666666666666666), -2.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -2.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

4. Applied egg-rr 49.1%

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

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x} + -1}{x} \cdot \frac{-1}{x} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x + -1}}{x} \cdot \frac{-1}{x} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\cos x + -1}}} \cdot \frac{-1}{x} \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{x}{\cos x + -1} \cdot x}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\frac{x}{\cos x + -1} \cdot x} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\cos x + -1} \cdot x}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{x}{\cos x + -1} \cdot x}} \]

   8. lower-/.f64 48.8

      \[\leadsto \frac{-1}{\color{blue}{\frac{x}{\cos x + -1}} \cdot x} \]

6. Applied egg-rr 48.8%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\cos x + -1} \cdot x}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} - 2}} \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{-1}{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{-1}{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-1}{x \cdot \left(\frac{-1}{6} \cdot x\right) + \color{blue}{-2}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, -2\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, -2\right)} \]

   8. lower-*.f64 78.5

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, -2\right)} \]

9. Simplified 78.5%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -2\right)}} \]
10. Add Preprocessing

Alternative 6: 63.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.35e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.3499999999999999e77

   1. Initial program 36.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 66.4%

      \[\leadsto \color{blue}{0.5} \]

   if 1.3499999999999999e77 < x

   1. Initial program 99.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
   4. Step-by-step derivation

   5. Simplified 65.0%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{x \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0}{\color{blue}{x \cdot x}} \]

      3. div0 65.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 27.3% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 48.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
4. Step-by-step derivation

5. Simplified 23.9%

   \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{x \cdot x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{0}{\color{blue}{x \cdot x}} \]

   3. div0 24.7

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 24.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))