cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 8.3s

Alternatives: 9

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

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

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 45.7%

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

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

   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 74.2

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

4. Applied rewrites 74.2%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (fma -0.041666666666666664 (* x x) 0.5)
   (* (- 1.0 (cos x)) (pow x -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 30.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

   4. Applied rewrites 99.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x} \cdot \left(\cos x - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (fma -0.041666666666666664 (* x x) 0.5)
   (* (/ (/ -1.0 x) x) (- (cos x) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x) / x) * Float64(cos(x) - 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0037], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0037000000000000002

   1. Initial program 30.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x} \cdot \left(\cos x - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 30.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

Alternative 5: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 30.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

Alternative 6: 64.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x} \cdot x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+38)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 (* (/ 1.0 x) x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+38) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - ((1.0 / x) * x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(1.0 / x) * x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+38], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e38

   1. Initial program 33.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if 6.5e38 < x

   1. Initial program 99.5%

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

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 55.1

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

   7. Applied rewrites 55.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x} \cdot x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+38)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+38) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+38], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 6.5e38

   1. Initial program 33.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if 6.5e38 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 54.8%

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

Alternative 8: 64.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8e+75) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 8e+75) {
		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 <= 8d+75) 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 <= 8e+75) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e+75:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e+75)
		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 <= 8e+75)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e+75], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999941e75

   1. Initial program 36.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.1%

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

   if 7.99999999999999941e75 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.5%

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

Alternative 9: 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 45.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.6%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))