cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.8%

Time: 9.0s

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: 50.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.5%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 74.7

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

4. Applied rewrites 74.7%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (fma -0.041666666666666664 (* x x) 0.5)
   (* (- 1.0 (cos x)) (pow x -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) * pow(x, -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) * (x ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \cos x - \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.006)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- (* (/ -1.0 x) (cos x)) (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.006) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = (((-1.0 / x) * cos(x)) - (-1.0 / x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.006)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 / x) * cos(x)) - Float64(-1.0 / x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.006], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(-1.0 / x), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0060000000000000001

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.0060000000000000001 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. div-sub N/A

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

      7. frac-2neg N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. distribute-neg-frac2 N/A

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

      20. lift-neg.f64 N/A

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

   6. Applied rewrites 99.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \cos x - \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{\frac{x}{\cos x - 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (/ -1.0 x) (/ x (- (cos x) 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = (-1.0 / x) / (x / (cos(x) - 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(-1.0 / x) / Float64(x / Float64(cos(x) - 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / N[(x / N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. mul-1-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lower-/.f64 N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

          \[\leadsto \frac{\frac{-1}{x}}{\frac{x}{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos x\right)\right)\right)}}} \]

      16. neg-mul-1 N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

          \[\leadsto \frac{\frac{-1}{x}}{\frac{x}{\color{blue}{\left(0 - -1 \cdot \cos x\right) - 1}}} \]

      19. neg-sub0 N/A

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

      20. neg-mul-1 N/A

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

      21. remove-double-neg N/A

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

      22. lower--.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 5: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (* (- (cos x) 1.0) (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = ((cos(x) - 1.0) * (-1.0 / x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(cos(x) - 1.0) * Float64(-1.0 / x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/r/ N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \color{blue}{\left(0 - \left(1 - \cos x\right)\right)}}{x} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \left(0 - \color{blue}{\left(1 - \cos x\right)}\right)}{x} \]

      13. sub-neg N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos x\right)\right)\right)}\right)}{x} \]

      14. neg-mul-1 N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \color{blue}{\left(\left(0 - -1 \cdot \cos x\right) - 1\right)}}{x} \]

      17. neg-sub0 N/A

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

      18. neg-mul-1 N/A

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

      19. remove-double-neg N/A

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

      20. lower--.f64 99.3

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

   8. Applied rewrites 99.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(-0.041666666666666664 * N[(x * x), $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 < 0.00419999999999999974

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.3%

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

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(-0.041666666666666664 * N[(x * x), $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 < 0.00419999999999999974

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.0%

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

Alternative 8: 64.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5e+76) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 4.5e+76) {
		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 <= 4.5d+76) 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 <= 4.5e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.5e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5e+76)
		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 <= 4.5e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.5e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999997e76

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 64.2%

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

   if 4.4999999999999997e76 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 56.6%

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

Alternative 9: 79.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

4. Applied rewrites 48.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{6} \cdot {x}^{2} + 2}} \]

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 76.9

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

7. Applied rewrites 76.9%

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

Alternative 10: 52.2% 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 47.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.5%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))