cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.8%

Time: 11.4s

Alternatives: 10

Speedup: 120.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 71.5

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

4. Applied rewrites 71.5%

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

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(-1 + \cos x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.088)
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (* (/ -1.0 x) (+ -1.0 (cos x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = ((-1.0 / x) * (-1.0 + cos(x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.088)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x) * Float64(-1.0 + cos(x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.088], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 0.087999999999999995 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.5

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

   6. Applied rewrites 99.5%

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

Alternative 3: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \frac{-1 + \cos x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.088)
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (* (/ -1.0 x) (/ (+ -1.0 (cos x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = (-1.0 / x) * ((-1.0 + cos(x)) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.088)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(-1.0 / x) * Float64(Float64(-1.0 + cos(x)) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.088], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] * N[(N[(-1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 0.087999999999999995 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 99.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \frac{-1 + \cos x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.088)
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.088)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -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, 0.088], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -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 < 0.087999999999999995

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 0.087999999999999995 < x

   1. Initial program 99.0%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower--.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 5: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.088)
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.088)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -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, 0.088], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -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 < 0.087999999999999995

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 0.087999999999999995 < 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 6: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x \cdot x}\\ \mathbf{if}\;x \leq 4.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \frac{1}{-x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (* x x))))
   (if (<= x 4.6)
     (fma
      (* x x)
      (fma
       x
       (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
       -0.041666666666666664)
      0.5)
     (fma t_0 t_0 (/ 1.0 (- (* x x)))))))

C

double code(double x) {
	double t_0 = x / (x * x);
	double tmp;
	if (x <= 4.6) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = fma(t_0, t_0, (1.0 / -(x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / Float64(x * x))
	tmp = 0.0
	if (x <= 4.6)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = fma(t_0, t_0, Float64(1.0 / Float64(-Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.6], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(1.0 / (-N[(x * x), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 4.5999999999999996 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-div N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{x}{x \cdot x}, \frac{1}{-x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, \frac{1}{x}, \frac{-1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6)
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (fma (/ 1.0 x) (/ 1.0 x) (/ -1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.6) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x), (1.0 / x), (-1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6)
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = fma(Float64(1.0 / x), Float64(1.0 / x), Float64(-1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.6], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 66.0%

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

   if 4.5999999999999996 < x

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{1}{x}, \color{blue}{\frac{-1}{{x}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

Alternative 8: 63.7% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.8e+38)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -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((x * x), 0.001388888888888889, -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(x * x), 0.001388888888888889, -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[(x * x), $MachinePrecision] * 0.001388888888888889 + -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 38.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if 6.79999999999999992e38 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.5%

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

Alternative 9: 63.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = fma(-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 <= 3.5)
		tmp = fma(-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, 3.5], N[(-0.041666666666666664 * 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 < 3.5

   1. Initial program 37.1%

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

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

   5. Applied rewrites 65.6%

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

   if 3.5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.5%

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

Alternative 10: 51.8% accurate, 120.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.0%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))