cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 11.8s

Alternatives: 9

Speedup: 17.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.9% 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

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (sin x) x) (/ x (tan (* x 0.5)))))

C

double code(double x) {
	return (sin(x) / x) / (x / tan((x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) / x) / (x / tan((x * 0.5d0)))
end function

Java

public static double code(double x) {
	return (Math.sin(x) / x) / (x / Math.tan((x * 0.5)));
}

Python

def code(x):
	return (math.sin(x) / x) / (x / math.tan((x * 0.5)))

Julia

function code(x)
	return Float64(Float64(sin(x) / x) / Float64(x / tan(Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) / x) / (x / tan((x * 0.5)));
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / N[(x / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. 1-sub-cos N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. hang-0p-tan N/A

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

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

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

   9. /-lowering-/.f64 78.6

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

4. Applied egg-rr 78.6%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   9. div-inv N/A

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

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

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

   11. metadata-eval 99.8

       \[\leadsto \frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]
7. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x} \cdot \tan \left(x \cdot 0.5\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (/ (sin x) x) (tan (* x 0.5))) x))

C

double code(double x) {
	return ((sin(x) / x) * tan((x * 0.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sin(x) / x) * tan((x * 0.5d0))) / x
end function

Java

public static double code(double x) {
	return ((Math.sin(x) / x) * Math.tan((x * 0.5))) / x;
}

Python

def code(x):
	return ((math.sin(x) / x) * math.tan((x * 0.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(sin(x) / x) * tan(Float64(x * 0.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((sin(x) / x) * tan((x * 0.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. 1-sub-cos N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. hang-0p-tan N/A

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

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

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

   9. /-lowering-/.f64 78.6

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

4. Applied egg-rr 78.6%

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

   1. associate-/r* N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. div-inv N/A

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

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

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

   9. metadata-eval N/A

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

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

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

   11. sin-lowering-sin.f64 99.7

       \[\leadsto \frac{\tan \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{\sin x}}{x}}{x} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x}}{x}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{\frac{\sin x}{x} \cdot \tan \left(x \cdot 0.5\right)}{x} \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \tan \left(x \cdot 0.5\right) \cdot \frac{\frac{\sin x}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (tan (* x 0.5)) (/ (/ (sin x) x) x)))

C

double code(double x) {
	return tan((x * 0.5)) * ((sin(x) / x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) * ((sin(x) / x) / x)
end function

Java

public static double code(double x) {
	return Math.tan((x * 0.5)) * ((Math.sin(x) / x) / x);
}

Python

def code(x):
	return math.tan((x * 0.5)) * ((math.sin(x) / x) / x)

Julia

function code(x)
	return Float64(tan(Float64(x * 0.5)) * Float64(Float64(sin(x) / x) / x))
end

MATLAB

function tmp = code(x)
	tmp = tan((x * 0.5)) * ((sin(x) / x) / x);
end

Wolfram

code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. 1-sub-cos N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. hang-0p-tan N/A

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

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

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

   9. /-lowering-/.f64 78.6

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

4. Applied egg-rr 78.6%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   9. div-inv N/A

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

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

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

   11. metadata-eval 99.8

       \[\leadsto \frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]
7. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 99.7

      \[\leadsto \frac{\frac{\sin x}{x} \cdot \tan \color{blue}{\left(x \cdot 0.5\right)}}{x} \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \tan \left(x \cdot 0.5\right)}{x}} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.6

      \[\leadsto \tan \left(x \cdot 0.5\right) \cdot \frac{\frac{\color{blue}{\sin x}}{x}}{x} \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right) \cdot \frac{\frac{\sin x}{x}}{x}} \]
11. Add Preprocessing

Alternative 4: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.102)
   (fma
    x
    (*
     x
     (fma
      x
      (* x (fma (* x x) -2.48015873015873e-5 0.001388888888888889))
      -0.041666666666666664))
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.102) {
		tmp = fma(x, (x * fma(x, (x * fma((x * x), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.102)
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(Float64(x * x), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.102], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $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 < 0.101999999999999993

   1. Initial program 36.4%

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

   4. Applied egg-rr 36.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   7. Simplified 66.8%

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

   if 0.101999999999999993 < x

   1. Initial program 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

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

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

      13. metadata-eval N/A

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

      14. cos-lowering-cos.f64 99.4

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

   6. Applied egg-rr 99.4%

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

Alternative 5: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.102)
   (fma
    x
    (*
     x
     (fma
      x
      (* x (fma (* x x) -2.48015873015873e-5 0.001388888888888889))
      -0.041666666666666664))
    0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.102) {
		tmp = fma(x, (x * fma(x, (x * fma((x * x), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.102)
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(Float64(x * x), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.102], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $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 < 0.101999999999999993

   1. Initial program 36.4%

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

   4. Applied egg-rr 36.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   7. Simplified 66.8%

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

   if 0.101999999999999993 < x

   1. Initial program 99.0%

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

Alternative 6: 78.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

4. Applied egg-rr 54.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} - 2}} \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. metadata-eval 76.7

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, \color{blue}{-2}\right)} \]

7. Simplified 76.7%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -2\right)}} \]
8. Add Preprocessing

Alternative 7: 62.3% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 36.7%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 66.3

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

   7. Simplified 66.3%

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

   if 3.5 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 54.3%

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

      1. metadata-eval N/A

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

      2. div0 54.3

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

   7. Applied egg-rr 54.3%

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

Alternative 8: 63.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.8e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 5.8e+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 <= 5.8d+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 <= 5.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.8e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.8e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.8000000000000003e76

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 61.9%

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

   if 5.8000000000000003e76 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.8%

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

      1. metadata-eval N/A

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

      2. div0 67.8

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

   7. Applied egg-rr 67.8%

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

Alternative 9: 27.3% accurate, 120.0× 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 54.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 27.1%

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

   1. metadata-eval N/A

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

   2. div0 27.7

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

7. Applied egg-rr 27.7%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))