cos2 (problem 3.4.1)

Percentage Accurate: 51.4% → 99.8%

Time: 12.0s

Alternatives: 15

Speedup: 4.6×

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: 51.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x\_m \cdot \tan \left(x\_m \cdot 0.5\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.05)
   (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 <= 0.05) {
		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 <= 0.05)
		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(sin(x_m) * tan(Float64(x_m * 0.5))) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.05], 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[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 0.050000000000000003 < x

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.4

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m \cdot \tan \left(x\_m \cdot 0.5\right)}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.05)
   (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 <= 0.05) {
		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 <= 0.05)
		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 * 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, 0.05], 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 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 0.050000000000000003 < x

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
   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. lower-/.f64 N/A

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

      5. lower-*.f64 99.6

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-5)
   (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 <= 5e-5) {
		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 <= 5e-5)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(tan(Float64(x_m * 0.5)) * 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, 5e-5], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Tan[N[(x$95$m * 0.5), $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 < 5.00000000000000024e-5

   1. Initial program 37.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 64.3

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

   5. Applied rewrites 64.3%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 99.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

4. Final simplification 74.9%

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

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.091:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x\_m}}{x\_m} \cdot \left(-1 + \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.091)
   (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 <= 0.091) {
		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 <= 0.091)
		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 / x_m) / 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, 0.091], 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 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(-1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.090999999999999998

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 0.090999999999999998 < x

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.091:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x} \cdot \left(-1 + \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.091:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.091)
   (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 <= 0.091) {
		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 <= 0.091)
		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, 0.091], 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 < 0.090999999999999998

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 0.090999999999999998 < x

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

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

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.091:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.091)
   (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 <= 0.091) {
		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 <= 0.091)
		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, 0.091], 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 < 0.090999999999999998

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 0.090999999999999998 < x

   1. Initial program 99.4%

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

Alternative 7: 76.0% accurate, 1.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x_m x_m))))
   (if (<= x_m 4.6)
     (/
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma
         x_m
         (* x_m (fma (* x_m x_m) -2.48015873015873e-5 0.001388888888888889))
         -0.041666666666666664)
        0.5))
      x_m)
     (fma (/ 1.0 x_m) (* x_m t_0) (- t_0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. 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 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 69.9

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 42.6%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
   8. 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. lft-mult-inverse N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 43.9

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

   9. Applied rewrites 43.9%

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

4. Final simplification 58.6%

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

Alternative 8: 76.0% accurate, 2.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.6)
   (/
    (*
     x_m
     (fma
      (* x_m x_m)
      (fma
       x_m
       (* x_m (fma (* x_m x_m) -2.48015873015873e-5 0.001388888888888889))
       -0.041666666666666664)
      0.5))
    x_m)
   (fma (/ 1.0 x_m) (/ x_m (* x_m x_m)) (- (/ 1.0 (* x_m x_m))))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.6)
		tmp = Float64(Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664), 0.5)) / x_m);
	else
		tmp = fma(Float64(1.0 / x_m), Float64(x_m / Float64(x_m * x_m)), Float64(-Float64(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, 4.6], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(x$95$m / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + (-N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. 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 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 69.9

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 42.6%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
   8. 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. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 43.8

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

   9. Applied rewrites 43.8%

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

4. Final simplification 58.6%

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

Alternative 9: 76.0% accurate, 2.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.6)
   (/
    (*
     x_m
     (fma
      (* x_m x_m)
      (fma
       x_m
       (* x_m (fma (* x_m x_m) -2.48015873015873e-5 0.001388888888888889))
       -0.041666666666666664)
      0.5))
    x_m)
   (/ (fma (- (/ -1.0 x_m)) x_m -1.0) (* x_m x_m))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.6)
		tmp = Float64(Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664), 0.5)) / x_m);
	else
		tmp = Float64(fma(Float64(-Float64(-1.0 / x_m)), x_m, -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, 4.6], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[((-N[(-1.0 / x$95$m), $MachinePrecision]) * x$95$m + -1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. 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 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 69.9

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 44.1%

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

4. Final simplification 58.7%

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

Alternative 10: 76.0% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.6)
   (fma
    (* x_m x_m)
    (fma
     x_m
     (* x_m (fma x_m (* x_m -2.48015873015873e-5) 0.001388888888888889))
     -0.041666666666666664)
    0.5)
   (/ (fma (- (/ -1.0 x_m)) x_m -1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.6) {
		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 = fma(-(-1.0 / x_m), x_m, -1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.6)
		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(fma(Float64(-Float64(-1.0 / x_m)), x_m, -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, 4.6], 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 / x$95$m), $MachinePrecision]) * x$95$m + -1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 44.1%

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

4. Final simplification 58.7%

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

Alternative 11: 75.8% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.6)
   (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 (/ 1.0 x_m))) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.6) {
		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 * (1.0 / x_m))) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.6)
		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 - Float64(x_m * Float64(1.0 / 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, 4.6], 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[(x$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 43.0%

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

Alternative 12: 75.6% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.6)
   (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 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.6) {
		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 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.6)
		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 - 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, 4.6], 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 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5999999999999996

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 64.9%

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

   if 4.5999999999999996 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 42.6%

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

Alternative 13: 75.7% accurate, 4.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 6.8e+38)
   (fma
    (* x_m x_m)
    (fma (* x_m x_m) 0.001388888888888889 -0.041666666666666664)
    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 <= 6.8e+38) {
		tmp = fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, -0.041666666666666664), 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 <= 6.8e+38)
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, -0.041666666666666664), 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, 6.8e+38], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $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 < 6.79999999999999992e38

   1. Initial program 41.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 61.5

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

   5. Applied rewrites 61.5%

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

   if 6.79999999999999992e38 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 49.3%

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

Alternative 14: 75.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.7e+76) 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 <= 1.7e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	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 <= 1.7d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.7e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.7e+76:
		tmp = 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 <= 1.7e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.7e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.7e+76], 0.5, 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 < 1.6999999999999999e76

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.9%

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

   if 1.6999999999999999e76 < 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 60.5%

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

Alternative 15: 51.0% accurate, 120.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 46.8%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))