cos2 (problem 3.4.1)

Percentage Accurate: 50.1% → 99.8%

Time: 9.6s

Alternatives: 10

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: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (/ (tan (/ x 2.0)) x) (/ (sin x) x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.1%

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

4. Applied egg-rr 76.1%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.088d0) then
        tmp = 0.5d0 + (x * (x * ((-0.041666666666666664d0) + ((x * x) * (0.001388888888888889d0 + ((x * x) * (-2.48015873015873d-5)))))))
    else
        tmp = ((1.0d0 / x) * (1.0d0 - cos(x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = ((1.0 / x) * (1.0 - Math.cos(x))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.088:
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))))
	else:
		tmp = ((1.0 / x) * (1.0 - math.cos(x))) / x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.088)
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	else
		tmp = ((1.0 / x) * (1.0 - cos(x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 39.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.7%

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

   4. Applied egg-rr 67.7%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)} - \frac{1}{24}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \frac{-1}{24}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \color{blue}{\left(\frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 63.4%

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

   if 0.087999999999999995 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. cos-lowering-cos.f64 99.2%

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

   5. Simplified 99.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. cos-lowering-cos.f64 99.2%

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

   7. Applied egg-rr 99.2%

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

Alternative 3: 75.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.088d0) then
        tmp = 0.5d0 + (x * (x * ((-0.041666666666666664d0) + ((x * x) * (0.001388888888888889d0 + ((x * x) * (-2.48015873015873d-5)))))))
    else
        tmp = ((-1.0d0) / x) * ((cos(x) + (-1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.088) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = (-1.0 / x) * ((Math.cos(x) + -1.0) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.088:
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))))
	else:
		tmp = (-1.0 / x) * ((math.cos(x) + -1.0) / x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.088)
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	else
		tmp = (-1.0 / x) * ((cos(x) + -1.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 39.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.7%

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

   4. Applied egg-rr 67.7%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)} - \frac{1}{24}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \frac{-1}{24}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \color{blue}{\left(\frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 63.4%

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

   if 0.087999999999999995 < x

   1. Initial program 98.6%

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

   4. Applied egg-rr 99.3%

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

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot \left(0.001388888888888889 + \left(x \cdot x\right) \cdot -2.48015873015873 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.1)
   (+
    0.5
    (*
     x
     (*
      x
      (+
       -0.041666666666666664
       (*
        (* x x)
        (+ 0.001388888888888889 (* (* x x) -2.48015873015873e-5)))))))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.1d0) then
        tmp = 0.5d0 + (x * (x * ((-0.041666666666666664d0) + ((x * x) * (0.001388888888888889d0 + ((x * x) * (-2.48015873015873d-5)))))))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.1:
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.1)
		tmp = Float64(0.5 + Float64(x * Float64(x * Float64(-0.041666666666666664 + Float64(Float64(x * x) * Float64(0.001388888888888889 + Float64(Float64(x * x) * -2.48015873015873e-5)))))));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.1)
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.1], N[(0.5 + N[(x * N[(x * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * N[(0.001388888888888889 + N[(N[(x * x), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 39.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)} - \frac{1}{24}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \frac{-1}{24}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \color{blue}{\left(\frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 63.6%

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

   if 0.10000000000000001 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. cos-lowering-cos.f64 99.3%

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

   5. Simplified 99.3%

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

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot \left(0.001388888888888889 + \left(x \cdot x\right) \cdot -2.48015873015873 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.1)
   (+
    0.5
    (*
     x
     (*
      x
      (+
       -0.041666666666666664
       (*
        (* x x)
        (+ 0.001388888888888889 (* (* x x) -2.48015873015873e-5)))))))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.1d0) then
        tmp = 0.5d0 + (x * (x * ((-0.041666666666666664d0) + ((x * x) * (0.001388888888888889d0 + ((x * x) * (-2.48015873015873d-5)))))))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.1:
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.1)
		tmp = Float64(0.5 + Float64(x * Float64(x * Float64(-0.041666666666666664 + Float64(Float64(x * x) * Float64(0.001388888888888889 + Float64(Float64(x * x) * -2.48015873015873e-5)))))));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.1)
		tmp = 0.5 + (x * (x * (-0.041666666666666664 + ((x * x) * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.1], N[(0.5 + N[(x * N[(x * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * N[(0.001388888888888889 + N[(N[(x * x), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 39.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)} - \frac{1}{24}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \frac{-1}{24}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{24} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720}} + \frac{-1}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \color{blue}{\left(\frac{-1}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{40320}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{720}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{40320}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 63.6%

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

   if 0.10000000000000001 < x

   1. Initial program 98.7%

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

Alternative 6: 63.3% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (+ 1.0 -1.0) (* x x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = (1.0d0 + (-1.0d0)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (1.0 + -1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (1.0 + -1.0) / (x * x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (1.0 + -1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + -1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 62.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 62.8%

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

   if 3.5 < x

   1. Initial program 98.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 50.4%

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

4. Final simplification 59.7%

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

Alternative 7: 63.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) (+ 0.5 (* -0.041666666666666664 (* x x))) 0.0))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 62.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 62.8%

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

   if 3.5 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. cos-lowering-cos.f64 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 50.4%

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

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{0}{x}}{x} \]

      2. div0 N/A

         \[\leadsto \frac{0}{x} \]

      3. div0 50.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 50.4%

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

Alternative 8: 78.7% accurate, 11.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (+ -2.0 (* x (* x -0.16666666666666666)))))

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0 / (-2.0 + (x * (x * -0.16666666666666666)));
}

Python

def code(x):
	return -1.0 / (-2.0 + (x * (x * -0.16666666666666666)))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.0 / N[(-2.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. cos-lowering-cos.f64 56.1%

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

5. Simplified 56.1%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. div-inv N/A

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

   6. clear-num N/A

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

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

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

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

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

   9. distribute-frac-neg2 N/A

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

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

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. remove-double-neg N/A

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

   15. metadata-eval N/A

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

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

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

   17. cos-lowering-cos.f64 55.5%

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

7. Applied egg-rr 55.5%

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

8. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 75.6%

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

10. Simplified 75.6%

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

Alternative 9: 63.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.0000000000000004e76

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 58.2%

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

   if 8.0000000000000004e76 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. cos-lowering-cos.f64 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.9%

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

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{0}{x}}{x} \]

      2. div0 N/A

         \[\leadsto \frac{0}{x} \]

      3. div0 66.9%

         \[\leadsto 0 \]

   10. Applied egg-rr 66.9%

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

Alternative 10: 27.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. cos-lowering-cos.f64 56.1%

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

5. Simplified 56.1%

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

6. Taylor expanded in x around 0

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

8. Simplified 27.8%

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

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{0}{x}}{x} \]

   2. div0 N/A

      \[\leadsto \frac{0}{x} \]

   3. div0 27.8%

      \[\leadsto 0 \]

10. Applied egg-rr 27.8%

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

Reproduce

herbie shell --seed 2024140 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))