cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.6%

Time: 12.9s

Alternatives: 9

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: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x\_m}, \cos x\_m, \frac{-1}{x\_m}\right)}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (fma (/ 1.0 x_m) (cos x_m) (/ -1.0 x_m)) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = fma((1.0 / x_m), cos(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 <= 0.0052)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(fma(Float64(1.0 / x_m), cos(x_m), Float64(-1.0 / x_m)) / Float64(-x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0052], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] * N[Cos[x$95$m], $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 34.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.9%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. cos-lowering-cos.f64 N/A

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

      19. neg-lowering-neg.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

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

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

      11. /-lowering-/.f64 99.7

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

   6. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0064:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x\_m} + \frac{\cos x\_m}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0064)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (+ (/ -1.0 x_m) (/ (cos x_m) x_m)) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0064) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = ((-1.0 / x_m) + (cos(x_m) / x_m)) / -x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0064)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x_m) + Float64(cos(x_m) / x_m)) / Float64(-x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0064], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] + N[(N[Cos[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00640000000000000031

   1. Initial program 34.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 0.00640000000000000031 < x

   1. Initial program 98.9%

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

      1. div-sub N/A

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

      2. associate-/r* N/A

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

      3. frac-sub N/A

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

      4. cube-unmult N/A

         \[\leadsto \frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}}} \]

      6. inv-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{-1}} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{{x}^{2}} - x \cdot \cos x}{{x}^{3}} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(-1 + 2\right)}} - x \cdot \cos x}{{x}^{3}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{{x}^{\color{blue}{1}} - x \cdot \cos x}{{x}^{3}} \]

      10. unpow1 N/A

          \[\leadsto \frac{\color{blue}{x} - x \cdot \cos x}{{x}^{3}} \]

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

          \[\leadsto \frac{\color{blue}{x - x \cdot \cos x}}{{x}^{3}} \]

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

          \[\leadsto \frac{x - \color{blue}{x \cdot \cos x}}{{x}^{3}} \]

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

          \[\leadsto \frac{x - x \cdot \color{blue}{\cos x}}{{x}^{3}} \]

      14. cube-unmult N/A

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

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

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

      16. *-lowering-*.f64 88.5

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

   4. Applied egg-rr 88.5%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. frac-sub N/A

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

      7. metadata-eval N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. mul-1-neg N/A

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

      11. div-inv N/A

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

      12. frac-2neg N/A

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

      13. remove-double-neg N/A

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

      14. sub-div N/A

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

      15. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0064:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} + \frac{\cos x}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (fma -0.041666666666666664 (* x_m x_m) 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 <= 0.0052) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 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 <= 0.0052)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 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, 0.0052], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $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 < 0.0051999999999999998

   1. Initial program 34.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.9%

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

      1. div-sub N/A

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

      2. associate-/r* N/A

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

      3. frac-sub N/A

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

      4. cube-unmult N/A

         \[\leadsto \frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}}} \]

      6. inv-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{-1}} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{{x}^{2}} - x \cdot \cos x}{{x}^{3}} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(-1 + 2\right)}} - x \cdot \cos x}{{x}^{3}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{{x}^{\color{blue}{1}} - x \cdot \cos x}{{x}^{3}} \]

      10. unpow1 N/A

          \[\leadsto \frac{\color{blue}{x} - x \cdot \cos x}{{x}^{3}} \]

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

          \[\leadsto \frac{\color{blue}{x - x \cdot \cos x}}{{x}^{3}} \]

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

          \[\leadsto \frac{x - \color{blue}{x \cdot \cos x}}{{x}^{3}} \]

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

          \[\leadsto \frac{x - x \cdot \color{blue}{\cos x}}{{x}^{3}} \]

      14. cube-unmult N/A

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

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

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

      16. *-lowering-*.f64 88.5

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

   4. Applied egg-rr 88.5%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. remove-double-neg N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. frac-sub N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. sub-div N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 99.7

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

   6. Applied egg-rr 99.7%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (fma -0.041666666666666664 (* x_m x_m) 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 <= 0.0052) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 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 <= 0.0052)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 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, 0.0052], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $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 < 0.0051999999999999998

   1. Initial program 34.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.9%

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

Alternative 5: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot x\_m\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.125, 0.25\right), 0.5\right), x\_m\right)} - \frac{-1}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.2)
   (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
      (fma
       (* x_m (* x_m x_m))
       (fma (* x_m x_m) (fma (* x_m x_m) 0.125 0.25) 0.5)
       x_m))
     (/ -1.0 x_m))
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.2) {
		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 / fma((x_m * (x_m * x_m)), fma((x_m * x_m), fma((x_m * x_m), 0.125, 0.25), 0.5), 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 <= 4.2)
		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 / fma(Float64(x_m * Float64(x_m * x_m)), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.125, 0.25), 0.5), x_m)) - Float64(-1.0 / x_m)) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.2], 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[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125 + 0.25), $MachinePrecision] + 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 66.6%

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

   if 4.20000000000000018 < x

   1. Initial program 99.0%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. cos-lowering-cos.f64 N/A

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

      19. neg-lowering-neg.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}{x}\right)}{-x} \]

   7. Simplified 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}{x}\right)}{-x} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}}}{-x} \]

   9. Applied egg-rr 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \color{blue}{\frac{-1}{\frac{x}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}}}{-x} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right)\right)}}}{\mathsf{neg}\left(x\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right) + 1\right)}}}{\mathsf{neg}\left(x\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right)\right) + x \cdot 1}}}{\mathsf{neg}\left(x\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right)} + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right) + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       5. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{{x}^{3}} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right) + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       6. *-rgt-identity N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{{x}^{3} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)\right) + \color{blue}{x}}}{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right), x\right)}}}{\mathsf{neg}\left(x\right)} \]

   12. Simplified 69.3%

       \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.125, 0.25\right), 0.5\right), x\right)}}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.125, 0.25\right), 0.5\right), x\right)} - \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(x\_m \cdot \left(x\_m \cdot x\_m\right), \mathsf{fma}\left(x\_m, x\_m \cdot 0.25, 0.5\right), x\_m\right)} - \frac{-1}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.2)
   (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 (fma (* x_m (* x_m x_m)) (fma x_m (* x_m 0.25) 0.5) x_m))
     (/ -1.0 x_m))
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.2) {
		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 / fma((x_m * (x_m * x_m)), fma(x_m, (x_m * 0.25), 0.5), 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 <= 4.2)
		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 / fma(Float64(x_m * Float64(x_m * x_m)), fma(x_m, Float64(x_m * 0.25), 0.5), x_m)) - Float64(-1.0 / x_m)) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.2], 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[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.25), $MachinePrecision] + 0.5), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 66.6%

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

   if 4.20000000000000018 < x

   1. Initial program 99.0%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. cos-lowering-cos.f64 N/A

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

      19. neg-lowering-neg.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}{x}\right)}{-x} \]

   7. Simplified 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}{x}\right)}{-x} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}}}{-x} \]

   9. Applied egg-rr 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \color{blue}{\frac{-1}{\frac{x}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}}}{-x} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right) + 1\right)}}}{\mathsf{neg}\left(x\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right) + x \cdot 1}}}{\mathsf{neg}\left(x\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right)} + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right) + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       5. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{{x}^{3}} \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right) + x \cdot 1}}{\mathsf{neg}\left(x\right)} \]

       6. *-rgt-identity N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{{x}^{3} \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot {x}^{2}\right) + \color{blue}{x}}}{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}}{\mathsf{neg}\left(x\right)} \]

       8. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \color{blue}{{x}^{2}}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

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

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{2}}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       11. unpow2 N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

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

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, \frac{1}{2} + \frac{1}{4} \cdot {x}^{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       13. +-commutative N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{4} \cdot {x}^{2} + \frac{1}{2}}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{4} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       15. associate-*r* N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\left(\frac{1}{4} \cdot x\right) \cdot x} + \frac{1}{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       16. *-commutative N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{x \cdot \left(\frac{1}{4} \cdot x\right)} + \frac{1}{2}, x\right)}}{\mathsf{neg}\left(x\right)} \]

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

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} \cdot x, \frac{1}{2}\right)}, x\right)}}{\mathsf{neg}\left(x\right)} \]

       18. *-commutative N/A

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{4}}, \frac{1}{2}\right), x\right)}}{\mathsf{neg}\left(x\right)} \]

       19. *-lowering-*.f64 69.3

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, \color{blue}{x \cdot 0.25}, 0.5\right), x\right)}}{-x} \]

   12. Simplified 69.3%

       \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.25, 0.5\right), x\right)}}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.25, 0.5\right), x\right)} - \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(0.5, x\_m \cdot \left(x\_m \cdot x\_m\right), x\_m\right)} - \frac{-1}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.2)
   (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 (fma 0.5 (* 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 <= 4.2) {
		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 / fma(0.5, (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 <= 4.2)
		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 / fma(0.5, Float64(x_m * Float64(x_m * x_m)), x_m)) - Float64(-1.0 / x_m)) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.2], 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[(0.5 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 66.6%

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

   if 4.20000000000000018 < x

   1. Initial program 99.0%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. cos-lowering-cos.f64 N/A

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

      19. neg-lowering-neg.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}{x}\right)}{-x} \]

   7. Simplified 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \left(-\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}{x}\right)}{-x} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 3.1

         \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.5, 1\right)}}}{-x} \]

   9. Applied egg-rr 3.1%

      \[\leadsto \frac{\frac{-1}{x} - \color{blue}{\frac{-1}{\frac{x}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}}}}{-x} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\frac{1}{2} \cdot \color{blue}{{x}^{3}} + x}}{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{3}, x\right)}}}{\mathsf{neg}\left(x\right)} \]

       8. cube-mult N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 69.3

           \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right)}}{-x} \]

   12. Simplified 69.3%

       \[\leadsto \frac{\frac{-1}{x} - \frac{-1}{\color{blue}{\mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right)}}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right)} - \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 8.5e+76) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 8.5d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 8.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 8.5e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 8.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 8.5e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999992e76

   1. Initial program 38.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.2%

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

   if 8.49999999999999992e76 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Simplified 78.2%

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

      1. metadata-eval N/A

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

      2. div0 78.2

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

   7. Applied egg-rr 78.2%

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

Alternative 9: 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 49.9%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. Simplified 27.7%

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

   1. metadata-eval N/A

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

   2. div0 28.6

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

7. Applied egg-rr 28.6%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))