2cos (problem 3.3.5)

?

Percentage Accurate: 38.4% → 99.4%
Time: 16.7s
Precision: binary64
Cost: 33600

?

\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := 0.5 \cdot \left(x + x\right)\\ -2 \cdot \left(t_0 \cdot \left(t_0 \cdot \cos t_1 + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin t_1\right)\right) \end{array} \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))) (t_1 (* 0.5 (+ x x))))
   (* -2.0 (* t_0 (+ (* t_0 (cos t_1)) (* (cos (* 0.5 eps)) (sin t_1)))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	double t_1 = 0.5 * (x + x);
	return -2.0 * (t_0 * ((t_0 * cos(t_1)) + (cos((0.5 * eps)) * sin(t_1))));
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function
↓
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin((0.5d0 * eps))
    t_1 = 0.5d0 * (x + x)
    code = (-2.0d0) * (t_0 * ((t_0 * cos(t_1)) + (cos((0.5d0 * eps)) * sin(t_1))))
end function

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}
↓
public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	double t_1 = 0.5 * (x + x);
	return -2.0 * (t_0 * ((t_0 * Math.cos(t_1)) + (Math.cos((0.5 * eps)) * Math.sin(t_1))));
}

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)
↓
def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	t_1 = 0.5 * (x + x)
	return -2.0 * (t_0 * ((t_0 * math.cos(t_1)) + (math.cos((0.5 * eps)) * math.sin(t_1))))

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	t_1 = Float64(0.5 * Float64(x + x))
	return Float64(-2.0 * Float64(t_0 * Float64(Float64(t_0 * cos(t_1)) + Float64(cos(Float64(0.5 * eps)) * sin(t_1)))))
end

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end
↓
function tmp = code(x, eps)
	t_0 = sin((0.5 * eps));
	t_1 = 0.5 * (x + x);
	tmp = -2.0 * (t_0 * ((t_0 * cos(t_1)) + (cos((0.5 * eps)) * sin(t_1))));
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision]}, N[(-2.0 * N[(t$95$0 * N[(N[(t$95$0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\
t_1 := 0.5 \cdot \left(x + x\right)\\
-2 \cdot \left(t_0 \cdot \left(t_0 \cdot \cos t_1 + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin t_1\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 12 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. Initial program 36.7%

    \[\cos \left(x + \varepsilon\right) - \cos x \]

  2. Applied egg-rr45.8%

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
    Step-by-step derivation

    [Start]36.7%

    \[ \cos \left(x + \varepsilon\right) - \cos x \]

    diff-cos [=>]45.8%

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

    div-inv [=>]45.8%

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

    metadata-eval [=>]45.8%

    \[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

    div-inv [=>]45.8%

    \[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

    +-commutative [=>]45.8%

    \[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

    metadata-eval [=>]45.8%

    \[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

  3. Simplified74.9%

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]
    Step-by-step derivation

    [Start]45.8%

    \[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    *-commutative [=>]45.8%

    \[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    +-commutative [<=]45.8%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    associate--l+ [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    +-inverses [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    distribute-lft-in [=>]74.9%

    \[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    metadata-eval [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

    *-commutative [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

    +-commutative [<=]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

  4. Applied egg-rr99.4%

    \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x + x\right) \cdot 0.5\right)\right)}\right) \]
    Step-by-step derivation

    [Start]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \]

    +-commutative [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(\varepsilon + x\right) + x\right)}\right)\right) \]

    associate-+r+ [<=]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

    distribute-rgt-in [=>]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5 + \left(x + x\right) \cdot 0.5\right)}\right) \]

    *-commutative [<=]74.9%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(\color{blue}{0.5 \cdot \varepsilon} + \left(x + x\right) \cdot 0.5\right)\right) \]

    sin-sum [=>]99.4%

    \[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x + x\right) \cdot 0.5\right)\right)}\right) \]

  5. Final simplification99.4%

    \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy99.4%
Cost33600
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := 0.5 \cdot \left(x + x\right)\\ -2 \cdot \left(t_0 \cdot \left(t_0 \cdot \cos t_1 + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin t_1\right)\right) \end{array} \]
Alternative 2
Accuracy99.2%
Cost32841
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.00017\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]
Alternative 3
Accuracy77.2%
Cost13768
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0035:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]
Alternative 4
Accuracy76.9%
Cost13632
\[-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right) \]
Alternative 5
Accuracy66.0%
Cost13516
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;-\frac{{\sin \varepsilon}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]
Alternative 6
Accuracy69.1%
Cost13513
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{-86} \lor \neg \left(x \leq 4.7 \cdot 10^{-7}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \]
Alternative 7
Accuracy66.2%
Cost13449
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-13} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-112}\right):\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]
Alternative 8
Accuracy66.0%
Cost13388
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]
Alternative 9
Accuracy66.1%
Cost6988
\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.75 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Accuracy47.3%
Cost6857
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.18\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]
Alternative 11
Accuracy21.7%
Cost320
\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]
Alternative 12
Accuracy12.8%
Cost64
\[0 \]

Reproduce?

herbie shell --seed 2023271 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))