cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.3%

Time: 9.7s

Alternatives: 8

Speedup: 107.0×

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: 50.6% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0002)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (/ (sin x) (* x x)) (tan (/ x 2.0)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0002) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (sin(x) / (x * x)) * tan((x / 2.0));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0002d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (sin(x) / (x * x)) * tan((x / 2.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0002) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (Math.sin(x) / (x * x)) * Math.tan((x / 2.0));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0002:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (math.sin(x) / (x * x)) * math.tan((x / 2.0))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0002)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(sin(x) / Float64(x * x)) * tan(Float64(x / 2.0)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0002)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (sin(x) / (x * x)) * tan((x / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0002], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 32.9%

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

   2. Taylor expanded in x around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

   4. Simplified 69.4%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 99.1%

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

      1. frac-2neg 99.1%

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

      2. div-inv 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.2%

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

      3. sqr-neg 49.2%

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

      4. sqrt-unprod 49.2%

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

      5. add-sqr-sqrt 49.2%

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

      6. flip-- 49.2%

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

      7. frac-times 49.2%

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

      8. metadata-eval 49.2%

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

      9. 1-sub-cos 49.2%

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

      10. pow2 49.2%

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

      11. add-sqr-sqrt 0.0%

          \[\leadsto \frac{{\sin x}^{2} \cdot 1}{\left(1 + \cos x\right) \cdot \color{blue}{\left(\sqrt{x \cdot \left(-x\right)} \cdot \sqrt{x \cdot \left(-x\right)}\right)}} \]

      12. sqrt-unprod 71.4%

          \[\leadsto \frac{{\sin x}^{2} \cdot 1}{\left(1 + \cos x\right) \cdot \color{blue}{\sqrt{\left(x \cdot \left(-x\right)\right) \cdot \left(x \cdot \left(-x\right)\right)}}} \]

      13. swap-sqr 71.4%

          \[\leadsto \frac{{\sin x}^{2} \cdot 1}{\left(1 + \cos x\right) \cdot \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)}}} \]

      14. sqr-neg 71.4%

          \[\leadsto \frac{{\sin x}^{2} \cdot 1}{\left(1 + \cos x\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}}} \]

      15. sqrt-unprod 98.6%

          \[\leadsto \frac{{\sin x}^{2} \cdot 1}{\left(1 + \cos x\right) \cdot \color{blue}{\left(\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}\right)}} \]

      16. add-sqr-sqrt 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. *-rgt-identity 98.6%

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

      2. unpow2 98.6%

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

      3. *-commutative 98.6%

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

      4. times-frac 98.8%

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

      5. hang-0p-tan 99.7%

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

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ (tan (* x 0.5)) (* x (/ x (sin x)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) / (x * (x / sin(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.9%

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

   1. flip-- 49.7%

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

   2. div-inv 49.6%

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

   3. metadata-eval 49.6%

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

   4. 1-sub-cos 76.5%

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

   5. pow2 76.5%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Applied egg-rr 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. associate-*r/ 76.5%

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

   4. *-rgt-identity 76.5%

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

   5. hang-0p-tan 76.9%

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

5. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sin x}}} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x} \]

   3. frac-times 99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \tan \left(x \cdot 0.5\right)}{\frac{x}{\sin x} \cdot x}} \]

   4. *-un-lft-identity 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (tan((x * 0.5d0)) / x) * (sin(x) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.9%

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

   1. flip-- 49.7%

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

   2. div-inv 49.6%

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

   3. metadata-eval 49.6%

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

   4. 1-sub-cos 76.5%

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

   5. pow2 76.5%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Applied egg-rr 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. associate-*r/ 76.5%

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

   4. *-rgt-identity 76.5%

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

   5. hang-0p-tan 76.9%

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

5. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (/ (- 1.0 (cos x)) x) (/ 1.0 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) 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

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) * (1.0 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		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

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0054], 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 < 0.0054000000000000003

   1. Initial program 32.9%

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

   2. Taylor expanded in x around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

   4. Simplified 69.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.1%

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

      1. associate-/r* 99.0%

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

      2. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{-x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (+ (cos x) -1.0) (- x)) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((cos(x) + -1.0) / -x) / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((cos(x) + (-1.0d0)) / -x) / x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((Math.cos(x) + -1.0) / -x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((math.cos(x) + -1.0) / -x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) / Float64(-x)) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((cos(x) + -1.0) / -x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0054], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 32.9%

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

   2. Taylor expanded in x around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

   4. Simplified 69.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.1%

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

      1. frac-2neg 99.1%

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

      2. div-inv 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in x around 0 98.9%

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

      1. unpow2 98.9%

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

      2. associate-/r* 99.0%

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

   6. Simplified 99.0%

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

      1. associate-*r/ 99.1%

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

      2. frac-2neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. un-div-inv 99.0%

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

      5. neg-sub0 99.0%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(1 - \cos x\right)}}{-x}}{x} \]

      6. metadata-eval 99.0%

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

      7. associate--r- 99.0%

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

      8. metadata-eval 99.0%

         \[\leadsto \frac{\frac{\left(\color{blue}{0} - 1\right) + \cos x}{-x}}{x} \]

      9. metadata-eval 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{-x}}{x}\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		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

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0054], 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 < 0.0054000000000000003

   1. Initial program 32.9%

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

   2. Taylor expanded in x around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

   4. Simplified 69.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 7: 52.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ (tan (* x 0.5)) x))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 49.9%

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

   1. flip-- 49.7%

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

   2. div-inv 49.6%

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

   3. metadata-eval 49.6%

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

   4. 1-sub-cos 76.5%

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

   5. pow2 76.5%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Applied egg-rr 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. associate-*r/ 76.5%

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

   4. *-rgt-identity 76.5%

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

   5. hang-0p-tan 76.9%

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

5. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sin x}}} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x} \]

   3. frac-times 99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \tan \left(x \cdot 0.5\right)}{\frac{x}{\sin x} \cdot x}} \]

   4. *-un-lft-identity 99.8%

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in x around 0 53.7%

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

11. Final simplification 53.7%

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

Alternative 8: 51.8% accurate, 107.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.5)

C

x = abs(x);
double code(double x) {
	return 0.5;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.5;
}

Python

x = abs(x)
def code(x):
	return 0.5

Julia

x = abs(x)
function code(x)
	return 0.5
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 0.5;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.5

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around 0 52.8%

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

3. Final simplification 52.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))