cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.7%

Time: 9.7s

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: 51.3% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.2%

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

4. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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 2: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} 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 <= 0.032d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    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 <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.032:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	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 <= 0.032)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $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 < 0.032000000000000001

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 68.2%

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

   5. Simplified 68.2%

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

   if 0.032000000000000001 < x

   1. Initial program 98.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. --lowering--.f64 N/A

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

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

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. cos-lowering-cos.f64 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/r* N/A

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

      2. sub-div N/A

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

      3. div-inv N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

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

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. fmm-def N/A

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

      11. div-inv N/A

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

      12. sub-div N/A

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

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

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

      14. --lowering--.f64 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 99.1%

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

   6. Applied egg-rr 99.1%

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

Alternative 3: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} 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 <= 0.032d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    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 <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.032:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	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 <= 0.032)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $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 < 0.032000000000000001

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 68.2%

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

   5. Simplified 68.2%

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

   if 0.032000000000000001 < x

   1. Initial program 98.3%

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

Alternative 4: 64.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + x \cdot \left(x \cdot \left(0.001388888888888889 + \left(x \cdot x\right) \cdot -2.48015873015873 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2)
   (+
    0.5
    (*
     (* x x)
     (+
      -0.041666666666666664
      (* x (* x (+ 0.001388888888888889 (* (* x x) -2.48015873015873e-5)))))))
   (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + (x * (x * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.2d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + (x * (x * (0.001388888888888889d0 + ((x * x) * (-2.48015873015873d-5)))))))
    else
        tmp = (2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + (x * (x * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.2:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + (x * (x * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))))
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(x * Float64(x * Float64(0.001388888888888889 + Float64(Float64(x * x) * -2.48015873015873e-5)))))));
	else
		tmp = Float64(Float64(2.0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.2)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + (x * (x * (0.001388888888888889 + ((x * x) * -2.48015873015873e-5))))));
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.2], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(x * N[(x * N[(0.001388888888888889 + N[(N[(x * x), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 67.9%

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

   5. Simplified 67.9%

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

   if 4.20000000000000018 < x

   1. Initial program 98.3%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 60.0%

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

   7. Simplified 60.0%

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

Alternative 5: 65.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(2.0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 68.1%

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

   5. Simplified 68.1%

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

   if 2.7000000000000002 < x

   1. Initial program 98.3%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 60.0%

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

   7. Simplified 60.0%

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

Alternative 6: 64.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (+ 0.5 (* (* x x) -0.041666666666666664)) (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(2.0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-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 67.7%

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

   5. Simplified 67.7%

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

   if 2.39999999999999991 < x

   1. Initial program 98.3%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 0.5 (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.5
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 0.5, N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.1%

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

   if 2 < x

   1. Initial program 98.3%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 60.0%

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

   7. Simplified 60.0%

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

Alternative 8: 65.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 0.5 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.5
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 0.5, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.1%

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

   if 2 < x

   1. Initial program 98.3%

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 59.9%

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

   7. Simplified 59.9%

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

Alternative 9: 63.6% 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 39.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 63.7%

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

   if 8.0000000000000004e76 < x

   1. Initial program 98.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.0%

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

      1. metadata-eval N/A

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

      2. div0 68.0%

         \[\leadsto 0 \]

   7. Applied egg-rr 68.0%

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

Alternative 10: 27.6% 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 50.2%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 24.7%

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

   1. metadata-eval N/A

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

   2. div0 25.4%

      \[\leadsto 0 \]

7. Applied egg-rr 25.4%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))