cos2 (problem 3.4.1)

?

Percentage Accurate: 8.0% → 99.3%
Time: 21.6s
Precision: binary64
Cost: 20292

?

\[\frac{1 - \cos x}{x \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.498:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= (/ (- 1.0 (cos x)) (* x x)) 0.498)
   (/ (* (sin x) (tan (/ x 2.0))) (* x x))
   (+
    0.5
    (+
     (* 0.001388888888888889 (pow x 4.0))
     (* x (* x -0.041666666666666664))))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	double tmp;
	if (((1.0 - cos(x)) / (x * x)) <= 0.498) {
		tmp = (sin(x) * tan((x / 2.0))) / (x * x);
	} else {
		tmp = 0.5 + ((0.001388888888888889 * pow(x, 4.0)) + (x * (x * -0.041666666666666664)));
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 - cos(x)) / (x * x)) <= 0.498d0) then
        tmp = (sin(x) * tan((x / 2.0d0))) / (x * x)
    else
        tmp = 0.5d0 + ((0.001388888888888889d0 * (x ** 4.0d0)) + (x * (x * (-0.041666666666666664d0))))
    end if
    code = tmp
end function

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}
↓
public static double code(double x) {
	double tmp;
	if (((1.0 - Math.cos(x)) / (x * x)) <= 0.498) {
		tmp = (Math.sin(x) * Math.tan((x / 2.0))) / (x * x);
	} else {
		tmp = 0.5 + ((0.001388888888888889 * Math.pow(x, 4.0)) + (x * (x * -0.041666666666666664)));
	}
	return tmp;
}

def code(x):
	return (1.0 - math.cos(x)) / (x * x)
↓
def code(x):
	tmp = 0
	if ((1.0 - math.cos(x)) / (x * x)) <= 0.498:
		tmp = (math.sin(x) * math.tan((x / 2.0))) / (x * x)
	else:
		tmp = 0.5 + ((0.001388888888888889 * math.pow(x, 4.0)) + (x * (x * -0.041666666666666664)))
	return tmp

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end
↓
function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 - cos(x)) / Float64(x * x)) <= 0.498)
		tmp = Float64(Float64(sin(x) * tan(Float64(x / 2.0))) / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(Float64(0.001388888888888889 * (x ^ 4.0)) + Float64(x * Float64(x * -0.041666666666666664))));
	end
	return tmp
end

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end
↓
function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 - cos(x)) / (x * x)) <= 0.498)
		tmp = (sin(x) * tan((x / 2.0))) / (x * x);
	else
		tmp = 0.5 + ((0.001388888888888889 * (x ^ 4.0)) + (x * (x * -0.041666666666666664)));
	end
	tmp_2 = tmp;
end

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[LessEqual[N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], 0.498], N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.498:\\
\;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 6 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x)) < 0.498

    1. Initial program 14.7%

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

    2. Applied egg-rr99.2%

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

      [Start]14.7%

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

      flip-- [=>]14.5%

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

      div-inv [=>]14.5%

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

      metadata-eval [=>]14.5%

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

      1-sub-cos [=>]99.2%

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

      pow2 [=>]99.2%

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

    3. Simplified99.5%

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

      [Start]99.2%

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

      unpow2 [=>]99.2%

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

      associate-*l* [=>]99.3%

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

      associate-*r/ [=>]99.2%

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

      *-rgt-identity [=>]99.2%

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

      hang-0p-tan [=>]99.5%

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

    if 0.498 < (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x))

    1. Initial program 2.3%

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

    2. Taylor expanded in x around 0 100.0%

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

    3. Simplified100.0%

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

      [Start]100.0%

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

      +-commutative [=>]100.0%

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

      fma-def [=>]100.0%

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

      *-commutative [=>]100.0%

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

      unpow2 [=>]100.0%

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

      associate-*l* [=>]100.0%

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

    4. Applied egg-rr100.0%

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

      [Start]100.0%

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

      fma-udef [=>]100.0%

      \[ 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification99.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.498:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.3%
Cost20292
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.498:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array} \]
Alternative 2
Accuracy97.8%
Cost7300
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0285:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \end{array} \]
Alternative 3
Accuracy97.5%
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]
Alternative 4
Accuracy97.5%
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]
Alternative 5
Accuracy94.8%
Cost448
\[0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664 \]
Alternative 6
Accuracy94.6%
Cost64
\[0.5 \]

Reproduce?

herbie shell --seed 2023277 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))