cos2 (problem 3.4.1)

Percentage Accurate: 50.1% → 99.7%

Time: 10.0s

Alternatives: 9

Speedup: 34.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* (/ (* (/ -1.0 x) (sin x)) x) (tan (/ x 2.0)))))

C

double code(double x) {
	return -((((-1.0 / x) * sin(x)) / x) * tan((x / 2.0)));
}

Fortran

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

Java

public static double code(double x) {
	return -((((-1.0 / x) * Math.sin(x)) / x) * Math.tan((x / 2.0)));
}

Python

def code(x):
	return -((((-1.0 / x) * math.sin(x)) / x) * math.tan((x / 2.0)))

Julia

function code(x)
	return Float64(-Float64(Float64(Float64(Float64(-1.0 / x) * sin(x)) / x) * tan(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = -((((-1.0 / x) * sin(x)) / x) * tan((x / 2.0)));
end

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. frac-2neg 53.6%

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

   2. div-inv 53.5%

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

   3. distribute-rgt-neg-in 53.5%

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

3. Applied egg-rr 53.5%

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

   1. distribute-lft-neg-out 53.5%

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

   2. associate-/r* 54.6%

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

5. Simplified 54.6%

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

   1. flip-- 54.4%

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

   2. frac-2neg 54.4%

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

   3. frac-times 55.2%

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

   4. metadata-eval 55.2%

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

   5. 1-sub-cos 78.2%

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

   6. neg-mul-1 78.2%

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

   7. div-inv 78.2%

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

   8. remove-double-neg 78.2%

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

7. Applied egg-rr 78.2%

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

   1. *-commutative 78.2%

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

   2. *-commutative 78.2%

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

   3. associate-*r* 99.5%

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

   4. times-frac 99.5%

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

   5. hang-0p-tan 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right) + {x}^{4} \cdot 0.001388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.026)
   (+
    (fma (* x x) -0.041666666666666664 0.5)
    (* (pow x 4.0) 0.001388888888888889))
   (* (/ (- 1.0 (cos x)) x) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.026) {
		tmp = fma((x * x), -0.041666666666666664, 0.5) + (pow(x, 4.0) * 0.001388888888888889);
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.026)
		tmp = Float64(fma(Float64(x * x), -0.041666666666666664, 0.5) + Float64((x ^ 4.0) * 0.001388888888888889));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) * Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.026], N[(N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0259999999999999988

   1. Initial program 36.4%

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

   2. Taylor expanded in x around 0 64.5%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 64.5%

         \[\leadsto \color{blue}{\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + 0.001388888888888889 \cdot {x}^{4}} \]

      2. +-commutative 64.5%

         \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot {x}^{2} + 0.5\right)} + 0.001388888888888889 \cdot {x}^{4} \]

      3. *-commutative 64.5%

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot -0.041666666666666664} + 0.5\right) + 0.001388888888888889 \cdot {x}^{4} \]

      4. fma-def 64.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, -0.041666666666666664, 0.5\right)} + 0.001388888888888889 \cdot {x}^{4} \]

      5. unpow2 64.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.041666666666666664, 0.5\right) + 0.001388888888888889 \cdot {x}^{4} \]

      6. *-commutative 64.5%

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right) + \color{blue}{{x}^{4} \cdot 0.001388888888888889} \]

   4. Simplified 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right) + {x}^{4} \cdot 0.001388888888888889} \]

   if 0.0259999999999999988 < x

   1. Initial program 97.6%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right) + {x}^{4} \cdot 0.001388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 3: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0047:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0047)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (/ (- 1.0 (cos x)) x) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0047) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0047d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0047) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0047:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) * (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0047)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0047)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0047], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00470000000000000018

   1. Initial program 36.4%

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

   2. Taylor expanded in x around 0 64.7%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 64.7%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 64.7%

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

   4. Simplified 64.7%

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

   if 0.00470000000000000018 < x

   1. Initial program 97.6%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0047:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0047:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0047)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0047) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} 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 <= 0.0047d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    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 <= 0.0047) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0047:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0047)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	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 <= 0.0047)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0047], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $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 < 0.00470000000000000018

   1. Initial program 36.4%

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

   2. Taylor expanded in x around 0 64.7%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 64.7%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 64.7%

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

   4. Simplified 64.7%

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

   if 0.00470000000000000018 < x

   1. Initial program 97.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0047:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 79.2% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. frac-2neg 53.6%

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

   2. div-inv 53.5%

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

   3. distribute-rgt-neg-in 53.5%

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

3. Applied egg-rr 53.5%

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

5. Applied egg-rr 54.2%

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

6. Taylor expanded in x around 0 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r/ 79.1%

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

   3. metadata-eval 79.1%

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

8. Simplified 79.1%

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

9. Final simplification 79.1%

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

Alternative 6: 64.9% accurate, 11.8× 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 36.7%

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

   2. Taylor expanded in x around 0 64.5%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 64.5%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 64.5%

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

   4. Simplified 64.5%

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

   if 3.2000000000000002 < x

   1. Initial program 97.6%

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

      1. frac-2neg 97.6%

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

      2. div-inv 97.5%

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

      3. distribute-rgt-neg-in 97.5%

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

   3. Applied egg-rr 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in x around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r/ 61.6%

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

      3. metadata-eval 61.6%

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

   8. Simplified 61.6%

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

   9. Taylor expanded in x around inf 61.7%

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

       1. unpow2 61.7%

          \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   11. Simplified 61.7%

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

4. Final simplification 63.7%

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

Alternative 7: 65.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.5) 0.5 (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		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 <= 3.5d0) 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 <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 3.5], 0.5, N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

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. Taylor expanded in x around 0 64.9%

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

   if 3.5 < x

   1. Initial program 97.6%

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

      1. frac-2neg 97.6%

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

      2. div-inv 97.5%

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

      3. distribute-rgt-neg-in 97.5%

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

   3. Applied egg-rr 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in x around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r/ 61.6%

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

      3. metadata-eval 61.6%

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

   8. Simplified 61.6%

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

   9. Taylor expanded in x around inf 61.7%

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

       1. unpow2 61.7%

          \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   11. Simplified 61.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 8: 64.1% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 9.8e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 9.8e+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 <= 9.8d+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 <= 9.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.8e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.8e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.80000000000000053e76

   1. Initial program 42.4%

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

   2. Taylor expanded in x around 0 59.5%

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

   if 9.80000000000000053e76 < x

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 76.5%

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

   3. Taylor expanded in x around 0 76.5%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 27.9% 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 53.6%

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

2. Taylor expanded in x around 0 31.3%

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

3. Taylor expanded in x around 0 32.0%

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

4. Final simplification 32.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))