cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 7.4s

Alternatives: 10

Speedup: 120.0×

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: 50.7% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (/ (tan (/ x 2.0)) x) (sin x)) x))

C

double code(double x) {
	return ((tan((x / 2.0)) / x) * sin(x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((tan((x / 2.0d0)) / x) * sin(x)) / x
end function

Java

public static double code(double x) {
	return ((Math.tan((x / 2.0)) / x) * Math.sin(x)) / x;
}

Python

def code(x):
	return ((math.tan((x / 2.0)) / x) * math.sin(x)) / x

Julia

function code(x)
	return Float64(Float64(Float64(tan(Float64(x / 2.0)) / x) * sin(x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((tan((x / 2.0)) / x) * sin(x)) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 75.1

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

4. Applied rewrites 75.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \cdot \frac{\frac{\sin x}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (tan (/ x 2.0)) (/ (/ (sin x) x) x)))

C

double code(double x) {
	return tan((x / 2.0)) * ((sin(x) / x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) * ((sin(x) / x) / x)
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) * ((Math.sin(x) / x) / x);
}

Python

def code(x):
	return math.tan((x / 2.0)) * ((math.sin(x) / x) / x)

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) * Float64(Float64(sin(x) / x) / x))
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) * ((sin(x) / x) / x);
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 75.1

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

4. Applied rewrites 75.1%

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

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. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. hang-0p-tan N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot \frac{\frac{\sin x}{x}}{x}} \]
8. Add Preprocessing

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.1)
   (fma
    (*
     x
     (fma
      (pow x 4.0)
      -2.48015873015873e-5
      (fma (* 0.001388888888888889 x) x -0.041666666666666664)))
    x
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = fma((x * fma(pow(x, 4.0), -2.48015873015873e-5, fma((0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.1)
		tmp = fma(Float64(x * fma((x ^ 4.0), -2.48015873015873e-5, fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.1], N[(N[(x * N[(N[Power[x, 4.0], $MachinePrecision] * -2.48015873015873e-5 + N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 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 < 0.10000000000000001

   1. Initial program 32.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 97.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 4: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		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 <= 0.005)
		tmp = fma(Float64(x * 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, 0.005], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.0050000000000000001

   1. Initial program 32.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 0.0050000000000000001 < x

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		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 <= 0.005)
		tmp = fma(Float64(x * 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, 0.005], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.0050000000000000001

   1. Initial program 32.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 0.0050000000000000001 < x

   1. Initial program 97.1%

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

Alternative 6: 66.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \frac{1}{x}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.8)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (+ 1.0 (* x (/ 1.0 x))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 + (x * (1.0 / x))) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.8)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(1.0 / x))) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.8], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 + N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   1. Initial program 32.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 2.7999999999999998 < x

   1. Initial program 97.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Applied rewrites 58.7%

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

   7. Applied rewrites 64.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \frac{1}{x}}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3500000000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \frac{1}{x}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3500000000.0)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (- 1.0 (* x (/ 1.0 x))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3500000000.0) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - (x * (1.0 / x))) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3500000000.0)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(x * Float64(1.0 / x))) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3500000000.0], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e9

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 3.5e9 < x

   1. Initial program 97.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Applied rewrites 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. frac-sub N/A

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

      11. unpow1 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow1 N/A

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

      14. pow2 N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 59.9%

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

Alternative 8: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.8e+38)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8e+38) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8e+38)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.8e+38], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.80000000000000013e38

   1. Initial program 33.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 5.80000000000000013e38 < x

   1. Initial program 97.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Applied rewrites 60.5%

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

Alternative 9: 64.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3e+77) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3e+77], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e77

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.0%

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

   if 1.3000000000000001e77 < x

   1. Initial program 97.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Applied rewrites 66.4%

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

Alternative 10: 51.8% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024333 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))