cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.3%

Time: 11.9s

Alternatives: 11

Speedup: 17.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.8% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x\_m \cdot 0.5\right) \cdot \frac{\sin x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1e-18) 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 <= 1e-18) {
		tmp = 0.5;
	} else {
		tmp = tan((x_m * 0.5)) * (sin(x_m) / (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 <= 1d-18) then
        tmp = 0.5d0
    else
        tmp = tan((x_m * 0.5d0)) * (sin(x_m) / (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 <= 1e-18) {
		tmp = 0.5;
	} else {
		tmp = Math.tan((x_m * 0.5)) * (Math.sin(x_m) / (x_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1e-18:
		tmp = 0.5
	else:
		tmp = math.tan((x_m * 0.5)) * (math.sin(x_m) / (x_m * x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1e-18)
		tmp = 0.5;
	else
		tmp = Float64(tan(Float64(x_m * 0.5)) * Float64(sin(x_m) / 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 <= 1e-18)
		tmp = 0.5;
	else
		tmp = tan((x_m * 0.5)) * (sin(x_m) / (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, 1e-18], 0.5, 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 < 1.0000000000000001e-18

   1. Initial program 39.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 63.2%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 98.0%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. metadata-eval 99.6

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x\_m \cdot \frac{\tan \left(x\_m \cdot 0.5\right)}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.001)
   (fma
    (* x_m x_m)
    (fma x_m (* x_m 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 <= 0.001) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 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 <= 0.001)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	else
		tmp = Float64(sin(x_m) * Float64(tan(Float64(x_m * 0.5)) / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.001], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-3

   1. Initial program 39.3%

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

   4. Applied egg-rr 39.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

   if 1e-3 < x

   1. Initial program 99.5%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 99.5

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

   6. Applied egg-rr 99.5%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. flip-- N/A

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

   2. associate-/l/ N/A

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

   3. metadata-eval N/A

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

   4. 1-sub-cos N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. hang-0p-tan N/A

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

   11. tan-lowering-tan.f64 N/A

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

   12. /-lowering-/.f64 78.8

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

4. Applied egg-rr 78.8%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. div-inv N/A

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

   8. *-lowering-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f64 N/A

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

   11. sin-lowering-sin.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.095)
   (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 <= 0.095) {
		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 <= 0.095)
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * 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, 0.095], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $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 < 0.095000000000000001

   1. Initial program 39.3%

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

   4. Applied egg-rr 39.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 62.9

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

   7. Simplified 62.9%

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

   if 0.095000000000000001 < x

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*l/ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. sub-neg N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. cos-lowering-cos.f64 99.5

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

   6. Applied egg-rr 99.5%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.095)
   (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 <= 0.095) {
		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 <= 0.095)
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(Float64(x_m * 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, 0.095], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $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 < 0.095000000000000001

   1. Initial program 39.3%

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

   4. Applied egg-rr 39.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 62.9

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

   7. Simplified 62.9%

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

   if 0.095000000000000001 < x

   1. Initial program 99.5%

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

Alternative 6: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 23000000:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x\_m}, x\_m, x\_m \cdot \frac{-1}{x\_m}\right)}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 23000000.0)
   (fma
    (* x_m x_m)
    (fma x_m (* x_m 0.001388888888888889) -0.041666666666666664)
    0.5)
   (/ (fma (/ 1.0 x_m) x_m (* 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 <= 23000000.0) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x_m), x_m, (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 <= 23000000.0)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	else
		tmp = Float64(fma(Float64(1.0 / x_m), x_m, 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, 23000000.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] * x$95$m + 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 < 2.3e7

   1. Initial program 39.9%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 62.7

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

   7. Simplified 62.7%

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

   if 2.3e7 < x

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 N/A

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

      17. cos-lowering-cos.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 55.0

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

   7. Simplified 55.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. frac-times N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 55.2

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

   9. Applied egg-rr 55.2%

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

       1. +-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

       4. frac-add N/A

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

       5. /-lowering-/.f64 N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 N/A

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

       12. *-lowering-*.f64 55.5

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

   11. Applied egg-rr 55.5%

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

Alternative 7: 76.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 35000000000000:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x\_m}, \frac{-1}{x\_m}, \frac{1}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 35000000000000.0)
   (fma
    (* x_m x_m)
    (fma x_m (* x_m 0.001388888888888889) -0.041666666666666664)
    0.5)
   (fma (/ 1.0 x_m) (/ -1.0 x_m) (/ 1.0 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 35000000000000.0) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x_m), (-1.0 / 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 <= 35000000000000.0)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	else
		tmp = fma(Float64(1.0 / x_m), Float64(-1.0 / x_m), 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, 35000000000000.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e13

   1. Initial program 40.5%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 62.1

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

   7. Simplified 62.1%

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

   if 3.5e13 < x

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 N/A

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

      17. cos-lowering-cos.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 56.5

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

   7. Simplified 56.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. frac-times N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 57.1

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

   9. Applied egg-rr 57.1%

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

Alternative 8: 75.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 470000000000:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x\_m \cdot \frac{-1}{x\_m}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 470000000000.0)
   (fma
    (* x_m x_m)
    (fma x_m (* x_m 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 <= 470000000000.0) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 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 <= 470000000000.0)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 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, 470000000000.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $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 < 4.7e11

   1. Initial program 40.5%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 62.1

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

   7. Simplified 62.1%

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

   if 4.7e11 < x

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 N/A

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

      17. cos-lowering-cos.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 56.5

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

   7. Simplified 56.5%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. frac-add N/A

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

      4. /-lowering-/.f64 N/A

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

      5. inv-pow N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{{x}^{\color{blue}{0}} + x \cdot \frac{-1}{x}}{x \cdot x} \]

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 56.4

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

   9. Applied egg-rr 56.4%

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

Alternative 9: 75.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 8.5e+38)
   (fma
    (* x_m x_m)
    (fma x_m (* x_m 0.001388888888888889) -0.041666666666666664)
    0.5)
   0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8.5e+38) {
		tmp = fma((x_m * x_m), fma(x_m, (x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8.5e+38)
		tmp = fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.001388888888888889), -0.041666666666666664), 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 8.5e+38], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999997e38

   1. Initial program 41.1%

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

   4. Applied egg-rr 41.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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. accelerator-lowering-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. *-lowering-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 61.5

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

   7. Simplified 61.5%

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

   if 8.4999999999999997e38 < x

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 N/A

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

      17. cos-lowering-cos.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.5

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

   7. Simplified 58.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{0} \]
   9. Step-by-step derivation

   10. Simplified 58.0%

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

Alternative 10: 75.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.02e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.02e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.02d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.02e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.02e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.02e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.02e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.02e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e77

   1. Initial program 43.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.3%

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

   if 1.02e77 < x

   1. Initial program 99.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 N/A

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

      17. cos-lowering-cos.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 68.4

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{0} \]
   9. Step-by-step derivation

   10. Simplified 68.3%

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

Alternative 11: 27.2% accurate, 120.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. neg-sub0 N/A

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

   16. --lowering--.f64 N/A

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

   17. cos-lowering-cos.f64 53.8

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

4. Applied egg-rr 53.8%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 30.4

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

7. Simplified 30.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{0} \]
9. Step-by-step derivation

10. Simplified 30.6%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))