cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 11.6s

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: 50.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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

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

   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. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 80.0

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

4. Applied rewrites 80.0%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 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 50.3%

   \[\frac{1 - \cos x}{x \cdot x} \]
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

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

   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. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 75.3

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

4. Applied rewrites 75.3%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))