cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.8%

Time: 10.3s

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: 50.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.0%

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

4. Applied egg-rr 76.0%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

   8. /-lowering-/.f64 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (/ 1.0 (/ x (- 1.0 (cos x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} 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 <= 0.033d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    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 <= 0.033) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 / (x / (1.0 - Math.cos(x)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.033:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 / (x / (1.0 - math.cos(x)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 / Float64(x / Float64(1.0 - cos(x)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.033)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 / (x / (1.0 - cos(x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], 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[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   1. Initial program 35.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.5%

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

   5. Simplified 66.5%

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

   if 0.033000000000000002 < 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%

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

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

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

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

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

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

      5. cos-lowering-cos.f64 99.2%

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

   7. Applied egg-rr 99.2%

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

Alternative 3: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (+
    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 <= 0.033) {
		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 <= 0.033d0) 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 <= 0.033) {
		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 <= 0.033:
		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 <= 0.033)
		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 <= 0.033)
		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, 0.033], 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 < 0.033000000000000002

   1. Initial program 35.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.5%

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

   5. Simplified 66.5%

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

   if 0.033000000000000002 < 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 4: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (+
    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 <= 0.033) {
		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 <= 0.033d0) 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 <= 0.033) {
		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 <= 0.033:
		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 <= 0.033)
		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 <= 0.033)
		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, 0.033], 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 < 0.033000000000000002

   1. Initial program 35.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.5%

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

   5. Simplified 66.5%

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

   if 0.033000000000000002 < 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 5: 64.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2) (+ 0.5 (* (* x x) -0.041666666666666664)) (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.2d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.2:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.2)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 35.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.2%

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

   5. Simplified 66.2%

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

   if 3.2000000000000002 < 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%

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

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

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

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

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

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

      5. cos-lowering-cos.f64 99.2%

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 63.6%

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

   10. Simplified 63.6%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}{x}}}}{x} \]

   11. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 63.6%

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

   13. Simplified 63.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 51.5%

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

5. Simplified 51.5%

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

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

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

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

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

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

   5. cos-lowering-cos.f64 51.5%

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

7. Applied egg-rr 51.5%

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

8. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 76.9%

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

10. Simplified 76.9%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}{x}}}}{x} \]
11. Step-by-step derivation

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

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

    2. clear-num N/A

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

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

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

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

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

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

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

    6. *-lowering-*.f64 77.1%

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

12. Applied egg-rr 77.1%

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

Alternative 7: 78.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x \cdot 0.16666666666666666 + \frac{2}{x}}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 51.5%

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

5. Simplified 51.5%

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

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

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

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

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

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

   5. cos-lowering-cos.f64 51.5%

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

7. Applied egg-rr 51.5%

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

8. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 76.9%

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

10. Simplified 76.9%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}{x}}}}{x} \]

11. Taylor expanded in x around 0

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

    1. metadata-eval N/A

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

    2. fma-undefine N/A

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

    3. lft-mult-inverse N/A

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

    4. fma-define N/A

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

    5. associate-*l* N/A

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

    6. distribute-rgt-in N/A

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

    7. +-commutative N/A

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

    8. *-commutative N/A

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

    9. associate-/l* N/A

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

    10. unpow2 N/A

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

    11. associate-/l* N/A

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

    12. *-inverses N/A

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

    13. *-rgt-identity N/A

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

    14. *-commutative N/A

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

    15. distribute-rgt-in N/A

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

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

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

    17. *-commutative N/A

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

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

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

    19. associate-*l* N/A

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

    20. unpow2 N/A

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

    21. associate-/r* N/A

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

13. Simplified 76.9%

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

Alternative 8: 64.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.4) 0.5 (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.4d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = 0.5
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = 0.5;
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], 0.5, N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.0%

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

   if 3.39999999999999991 < 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%

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

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

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

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

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

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

      5. cos-lowering-cos.f64 99.2%

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 63.6%

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

   10. Simplified 63.6%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}{x}}}}{x} \]

   11. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 63.6%

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

   13. Simplified 63.6%

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

Alternative 9: 63.7% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.65e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.65e+77) {
		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 <= 1.65d+77) 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 <= 1.65e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.65e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999e77

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 62.2%

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

   if 1.6499999999999999e77 < x

   1. Initial program 98.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. associate-/r* N/A

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

      3. frac-sub N/A

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

      4. pow3 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x \cdot x\right) \cdot \frac{\cos x}{x}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 \cdot x\right), \left(\left(x \cdot x\right) \cdot \frac{\cos x}{x}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \left(\left(x \cdot x\right) \cdot \frac{\cos x}{x}\right)\right), \left({x}^{3}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{\cos x}{x}\right)\right)\right), \left({x}^{3}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\cos x}{x}\right)\right)\right), \left({x}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\cos x, x\right)\right)\right), \left({x}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{3}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{\left(1 + \color{blue}{2}\right)}\right)\right) \]

      13. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{1} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      14. unpow1 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(x \cdot {\color{blue}{x}}^{2}\right)\right) \]

      15. pow2 N/A

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

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

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

      17. *-lowering-*.f64 7.7%

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

   4. Applied egg-rr 7.7%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 1.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 80.0%

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

Alternative 10: 27.8% 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 50.4%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. associate-/r* N/A

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

   3. frac-sub N/A

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

   4. pow3 N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot x - \left(x \cdot x\right) \cdot \frac{\cos x}{x}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 \cdot x\right), \left(\left(x \cdot x\right) \cdot \frac{\cos x}{x}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \left(\left(x \cdot x\right) \cdot \frac{\cos x}{x}\right)\right), \left({x}^{3}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{\cos x}{x}\right)\right)\right), \left({x}^{3}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\cos x}{x}\right)\right)\right), \left({x}^{3}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\cos x, x\right)\right)\right), \left({x}^{3}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{3}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{\left(1 + \color{blue}{2}\right)}\right)\right) \]

   13. pow-prod-up N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left({x}^{1} \cdot \color{blue}{{x}^{2}}\right)\right) \]

   14. unpow1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), x\right)\right)\right), \left(x \cdot {\color{blue}{x}}^{2}\right)\right) \]

   15. pow2 N/A

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

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

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

   17. *-lowering-*.f64 21.4%

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

4. Applied egg-rr 21.4%

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

5. Taylor expanded in x around 0

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

7. Simplified 2.8%

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

8. Taylor expanded in x around 0

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

10. Simplified 25.6%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))