cos2 (problem 3.4.1)

Percentage Accurate: 51.5% → 99.8%

Time: 11.2s

Alternatives: 13

Speedup: 17.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0003:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \left(x\_m \cdot 0.5\right) \cdot \sin x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0003)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (/ (* (tan (* x_m 0.5)) (sin x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0003) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = ((tan((x_m * 0.5)) * sin(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0003)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(tan(Float64(x_m * 0.5)) * sin(x_m)) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0003], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999974e-4

   1. Initial program 36.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

   if 2.99999999999999974e-4 < x

   1. Initial program 98.2%

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

      1. lift-cos.f64 N/A

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

      2. flip-- N/A

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

      3. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.6

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \left(x \cdot 0.5\right) \cdot \sin x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (* (sin x_m) (tan (/ x_m 2.0))) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = (sin(x_m) * tan((x_m / 2.0))) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(sin(x_m) * tan(Float64(x_m / 2.0))) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.02], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.0200000000000000004 < x

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. 1-sub-cos N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. hang-0p-tan N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x\_m}{2}\right) \cdot \frac{\sin x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (* (tan (/ x_m 2.0)) (/ (sin x_m) (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = tan((x_m / 2.0)) * (sin(x_m) / (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = Float64(tan(Float64(x_m / 2.0)) * Float64(sin(x_m) / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.02], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.0200000000000000004 < x

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. flip-- N/A

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

      3. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{x\_m} \cdot \left(\tan \left(x\_m \cdot 0.5\right) \cdot \frac{\sin x\_m}{x\_m}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (/ 1.0 x_m) (* (tan (* x_m 0.5)) (/ (sin x_m) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / x_m) * (tan((x_m * 0.5)) * (sin(x_m) / x_m));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (1.0d0 / x_m) * (tan((x_m * 0.5d0)) * (sin(x_m) / x_m))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / x_m) * (Math.tan((x_m * 0.5)) * (Math.sin(x_m) / x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / x_m) * (math.tan((x_m * 0.5)) * (math.sin(x_m) / x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / x_m) * Float64(tan(Float64(x_m * 0.5)) * Float64(sin(x_m) / x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / x_m) * (tan((x_m * 0.5)) * (sin(x_m) / x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. lift-cos.f64 N/A

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

   2. flip-- N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 80.1

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

4. Applied rewrites 80.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 80.3

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 80.3

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

6. Applied rewrites 80.3%

   \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}}{x}} \]
7. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 80.2

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \]

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.6

       \[\leadsto \frac{1}{x} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x}\right)} \]
9. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x\_m \cdot \frac{\tan \left(x\_m \cdot 0.5\right)}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (* (sin x_m) (/ (tan (* x_m 0.5)) (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * fma(x_m, (x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	} else {
		tmp = sin(x_m) * (tan((x_m * 0.5)) / (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * -2.48015873015873e-5), 0.001388888888888889)), -0.041666666666666664), 0.5);
	else
		tmp = Float64(sin(x_m) * Float64(tan(Float64(x_m * 0.5)) / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.02], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.0200000000000000004 < x

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. flip-- N/A

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

      3. lift-*.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.5

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 0.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.088)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (/ -1.0 x_m) (/ x_m (+ -1.0 (cos x_m))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.088], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(-1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.087999999999999995 < x

   1. Initial program 99.3%

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

   4. Applied rewrites 99.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

4. Final simplification 76.2%

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

Alternative 7: 99.7% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.088)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.088], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.087999999999999995 < x

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-/r* N/A

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

      11. frac-2neg N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. neg-mul-1 N/A

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

      7. neg-mul-1 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-div N/A

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

   6. 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 8: 99.2% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.088)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.088], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.4%

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

   if 0.087999999999999995 < x

   1. Initial program 99.3%

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

Alternative 9: 75.9% accurate, 4.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (+ 1.0 -1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = (1.0 + -1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(1.0 + -1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 + -1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 37.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 3.5 < 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 52.3%

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

4. Final simplification 62.9%

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

Alternative 10: 78.6% accurate, 5.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (fma x_m (* x_m 0.16666666666666666) 2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(x$95$m * N[(x$95$m * 0.16666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 29.4%

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

7. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)}}} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 76.3

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

10. Applied rewrites 76.3%

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

Alternative 11: 75.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5) (fma -0.041666666666666664 (* x_m x_m) 0.5) 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 37.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 3.5 < 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 52.3%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. div0 52.3

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 52.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 8.2e+76) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8.2e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 8.2d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 8.2e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 8.2e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8.2e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 8.2e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 8.2e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999997e76

   1. Initial program 43.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.6%

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

   if 8.1999999999999997e76 < 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 72.0%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. div0 72.0

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 72.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 28.2% accurate, 120.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 54.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 26.0%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. div0 26.5

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 26.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))