cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.4%

Time: 13.9s

Alternatives: 9

Speedup: 17.8×

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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x\_m \cdot \left(1 - \cos x\_m\right)}{x\_m}}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014) 0.5 (/ (/ (/ (* x_m (- 1.0 (cos x_m))) x_m) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = (((x_m * (1.0 - cos(x_m))) / 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 <= 0.00014d0) then
        tmp = 0.5d0
    else
        tmp = (((x_m * (1.0d0 - cos(x_m))) / 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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = (((x_m * (1.0 - Math.cos(x_m))) / x_m) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.00014:
		tmp = 0.5
	else:
		tmp = (((x_m * (1.0 - math.cos(x_m))) / x_m) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00014)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(Float64(x_m * Float64(1.0 - cos(x_m))) / x_m) / 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 <= 0.00014)
		tmp = 0.5;
	else
		tmp = (((x_m * (1.0 - cos(x_m))) / 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, 0.00014], 0.5, N[(N[(N[(N[(x$95$m * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 98.0%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(x\right)} - \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\mathsf{neg}\left(x\right)}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(x\right)}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{x}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\cos x, x\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(0 - \color{blue}{x}\right)\right) \]

      20. --lowering--.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{x}\right)\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x} - \left(-\frac{\cos x}{x}\right)}{0 - x}} \]
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. mul-1-neg N/A

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

      4. sub0-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)} - \frac{\mathsf{neg}\left(\frac{\cos x}{x}\right)}{0 - x} \]

      5. frac-2neg N/A

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

      6. sub0-neg N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x} - \frac{\cos x}{x}\right), \color{blue}{x}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{x} - \frac{\cos x}{x}\right), x\right) \]

      11. sub-div N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) - \cos x}{x}\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) - \cos x\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \cos x\right), x\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      15. cos-lowering-cos.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   6. Applied egg-rr 99.3%

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

      1. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x} - \frac{\cos x}{x}\right), x\right) \]

      2. frac-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot x - x \cdot \cos x}{x \cdot x}\right), x\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot x}{x \cdot x} - \frac{x \cdot \cos x}{x \cdot x}\right), x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{x \cdot x} - \frac{x \cdot \cos x}{x \cdot x}\right), x\right) \]

      5. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x - x \cdot \cos x}{x \cdot x}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x - x \cdot \cos x}{x}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x - x \cdot \cos x}{x}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - x \cdot \cos x\right), x\right), x\right), x\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot x - x \cdot \cos x\right), x\right), x\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot x - \cos x \cdot x\right), x\right), x\right), x\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \left(1 - \cos x\right)\right), x\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - \cos x\right)\right), x\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \cos x\right)\right), x\right), x\right), x\right) \]

      14. cos-lowering-cos.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right)\right), x\right), x\right), x\right) \]

   8. Applied egg-rr 99.3%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{\sin x\_m}{x\_m}}{x\_m} \cdot \tan \left(\frac{x\_m}{2}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (/ (/ (sin x_m) x_m) x_m) (tan (/ x_m 2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return ((Math.sin(x_m) / x_m) / x_m) * Math.tan((x_m / 2.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return ((math.sin(x_m) / x_m) / x_m) * math.tan((x_m / 2.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(Float64(sin(x_m) / x_m) / x_m) * tan(Float64(x_m / 2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0));
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. flip-- N/A

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

   2. associate-/l/ N/A

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

   3. metadata-eval N/A

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

   4. 1-sub-cos N/A

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

   5. times-frac N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x}{x \cdot x}\right), \color{blue}{\left(\frac{\sin x}{1 + \cos x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin x, \left(x \cdot x\right)\right), \left(\frac{\color{blue}{\sin x}}{1 + \cos x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \left(x \cdot x\right)\right), \left(\frac{\sin \color{blue}{x}}{1 + \cos x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\sin x}{1 + \cos x}\right)\right) \]

   10. hang-0p-tan N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \tan \left(\frac{x}{2}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right) \]

   12. /-lowering-/.f64 71.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right) \]

4. Applied egg-rr 71.4%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\sin x}{x}}{x}\right), \mathsf{tan.f64}\left(\color{blue}{\mathsf{/.f64}\left(x, 2\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\sin x}{x}\right), x\right), \mathsf{tan.f64}\left(\color{blue}{\mathsf{/.f64}\left(x, 2\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin x, x\right), x\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, 2\right)\right)\right) \]

   4. sin-lowering-sin.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), x\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014) 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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - cos(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 <= 0.00014d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 - cos(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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.00014:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00014)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.00014)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(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, 0.00014], 0.5, 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 < 1.3999999999999999e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 98.0%

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

      1. div-sub N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(x\right)} - \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\mathsf{neg}\left(x\right)}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(x\right)}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{x}\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\mathsf{neg}\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\left(\frac{\cos x}{x}\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\cos x, x\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(\mathsf{neg}\left(x\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(0 - \color{blue}{x}\right)\right) \]

      20. --lowering--.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{x}\right)\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x} - \left(-\frac{\cos x}{x}\right)}{0 - x}} \]
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. mul-1-neg N/A

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

      4. sub0-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)} - \frac{\mathsf{neg}\left(\frac{\cos x}{x}\right)}{0 - x} \]

      5. frac-2neg N/A

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

      6. sub0-neg N/A

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

      7. frac-2neg N/A

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

      8. sub-div N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x} - \frac{\cos x}{x}\right), \color{blue}{x}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{x} - \frac{\cos x}{x}\right), x\right) \]

      11. sub-div N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) - \cos x}{x}\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) - \cos x\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \cos x\right), x\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      15. cos-lowering-cos.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   6. Applied egg-rr 99.3%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014) 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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(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 <= 0.00014d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.00014:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00014)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(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 <= 0.00014)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(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, 0.00014], 0.5, 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 < 1.3999999999999999e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 98.0%

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

Alternative 5: 78.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-1}{x\_m}}{\frac{-2}{x\_m} + x\_m \cdot -0.16666666666666666} \end{array} \]

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((-1.0d0) / x_m) / (((-2.0d0) / x_m) + (x_m * (-0.16666666666666666d0)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-1.0 / x_m) / ((-2.0 / x_m) + (x_m * -0.16666666666666666));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-1.0 / x_m) / ((-2.0 / x_m) + (x_m * -0.16666666666666666))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (-1.0 / x_m) / ((-2.0 / x_m) + (x_m * -0.16666666666666666));
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

4. Applied egg-rr 51.0%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), \color{blue}{\left(\frac{x}{\cos x + -1}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\color{blue}{x}}{\cos x + -1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\cos x + -1\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \left(-1 + \color{blue}{\cos x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\cos x}\right)\right)\right) \]

   10. cos-lowering-cos.f64 52.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{cos.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 52.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \color{blue}{\left(\frac{\frac{-1}{6} \cdot {x}^{2} - 2}{x}\right)}\right) \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} - \color{blue}{\frac{2}{x}}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2}}{x}}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \frac{x \cdot x}{x}\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(\frac{x}{x} \cdot \color{blue}{x}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(1 \cdot x\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{x}\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{-2}{x}\right), \left(\frac{-1}{6} \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   14. *-lowering-*.f64 79.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

9. Simplified 79.8%

   \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{\frac{-2}{x} + x \cdot -0.16666666666666666}} \]
10. Add Preprocessing

Alternative 6: 78.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (-1.0d0) / (x_m * (((-2.0d0) / x_m) + (x_m * (-0.16666666666666666d0))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return -1.0 / (x_m * ((-2.0 / x_m) + (x_m * -0.16666666666666666)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return -1.0 / (x_m * ((-2.0 / x_m) + (x_m * -0.16666666666666666)))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = -1.0 / (x_m * ((-2.0 / x_m) + (x_m * -0.16666666666666666)));
end

Wolfram

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

Derivation

1. Initial program 51.1%

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

4. Applied egg-rr 51.0%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), \color{blue}{\left(\frac{x}{\cos x + -1}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\color{blue}{x}}{\cos x + -1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\cos x + -1\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \left(-1 + \color{blue}{\cos x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\cos x}\right)\right)\right) \]

   10. cos-lowering-cos.f64 52.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{cos.f64}\left(x\right)\right)\right)\right) \]

6. Applied egg-rr 52.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \color{blue}{\left(\frac{\frac{-1}{6} \cdot {x}^{2} - 2}{x}\right)}\right) \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} - \color{blue}{\frac{2}{x}}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2}}{x}}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \frac{x \cdot x}{x}\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(\frac{x}{x} \cdot \color{blue}{x}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(1 \cdot x\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{x}\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{-2}{x}\right), \left(\frac{-1}{6} \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   14. *-lowering-*.f64 79.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

9. Simplified 79.8%

   \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{\frac{-2}{x} + x \cdot -0.16666666666666666}} \]

11. Applied egg-rr 79.8%

    \[\leadsto \color{blue}{\frac{-1}{x \cdot \left(\frac{-2}{x} + x \cdot -0.16666666666666666\right)}} \]
12. Add Preprocessing

Alternative 7: 78.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 3.5) 0.5 (/ 6.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 3.5d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5
	else:
		tmp = 6.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = 6.0 / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], 0.5, N[(6.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 34.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 3.5 < x

   1. Initial program 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), \color{blue}{\left(\frac{x}{\cos x + -1}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\color{blue}{x}}{\cos x + -1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\cos x + -1\right)}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \left(-1 + \color{blue}{\cos x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\cos x}\right)\right)\right) \]

      10. cos-lowering-cos.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{cos.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \color{blue}{\left(\frac{\frac{-1}{6} \cdot {x}^{2} - 2}{x}\right)}\right) \]
   8. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} - \color{blue}{\frac{2}{x}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2}}{x}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \frac{x \cdot x}{x}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(\frac{x}{x} \cdot \color{blue}{x}\right)\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot \left(1 \cdot x\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{-1}{6} \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{x}\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\left(\frac{-2}{x}\right), \left(\frac{-1}{6} \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      14. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(-2, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

   9. Simplified 52.9%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{\frac{-2}{x} + x \cdot -0.16666666666666666}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 53.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 53.1%

       \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.4e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.4e+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 <= 1.4d+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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, 1.4e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e77

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 60.5%

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

   if 1.4e77 < x

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{1}\right), \mathsf{*.f64}\left(x, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.7%

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

      1. metadata-eval N/A

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

      2. div0 69.7%

         \[\leadsto 0 \]

   7. Applied egg-rr 69.7%

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

Alternative 9: 27.8% accurate, 107.0× 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 51.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{1}\right), \mathsf{*.f64}\left(x, x\right)\right) \]
4. Step-by-step derivation

5. Simplified 29.1%

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

   1. metadata-eval N/A

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

   2. div0 30.0%

      \[\leadsto 0 \]

7. Applied egg-rr 30.0%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))