cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 99.8%

Time: 6.2s

Alternatives: 8

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: 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: 99.8% accurate, 0.5× speedup

Math

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

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(tan(Float64(x / 2.0)) / x) * Float64(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[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.7

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

4. Applied rewrites 76.7%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
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. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

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(Float64(Float64(tan(Float64(x * 0.5)) / x) / x) * sin(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.7

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

4. Applied rewrites 76.7%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
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. lower-*.f64 76.7

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-cos.f64 N/A

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

   15. hang-0p-tan N/A

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

   16. lower-tan.f64 N/A

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

   17. lower-/.f64 77.1

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

6. Applied rewrites 77.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-/.f64 N/A

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

   7. *-rgt-identity N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 99.7

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

8. Applied rewrites 99.7%

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

Alternative 3: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\cos x\right) + 1\right) \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (* (+ (- (cos x)) 1.0) (pow x -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (-cos(x) + 1.0) * pow(x, -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.028)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(-cos(x)) + 1.0) * (x ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.028], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[((-N[Cos[x], $MachinePrecision]) + 1.0), $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0280000000000000006

   1. Initial program 35.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 65.6%

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

   if 0.0280000000000000006 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 98.2

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

   4. Applied rewrites 98.2%

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

   6. Applied rewrites 99.1%

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

Alternative 4: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = fma((x * x), fma((0.001388888888888889 * 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 <= 0.028)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * 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, 0.028], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * 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 < 0.0280000000000000006

   1. Initial program 35.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 65.6%

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

   if 0.0280000000000000006 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 5: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = fma((x * x), fma((0.001388888888888889 * 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 <= 0.028)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * 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, 0.028], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * 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 < 0.0280000000000000006

   1. Initial program 35.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 65.6%

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

   if 0.0280000000000000006 < x

   1. Initial program 98.4%

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

Alternative 6: 62.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.8e+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 <= 6.8e+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 <= 6.8e+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, 6.8e+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 < 6.79999999999999992e38

   1. Initial program 39.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 62.3%

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

   if 6.79999999999999992e38 < x

   1. Initial program 98.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 54.5%

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

Alternative 7: 62.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.45)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = fma((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 <= 3.45)
		tmp = fma(Float64(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, 3.45], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 3.4500000000000002

   1. Initial program 35.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   if 3.4500000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 46.5%

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

Alternative 8: 51.4% 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 52.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.9%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))