Average Error: 39.7 → 0.5
Time: 8.0s
Precision: binary64
Cost: 33218
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001675426203356274:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015167359697961248:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0001675426203356274:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \leq 0.00015167359697961248:\\
\;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0001675426203356274)
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (if (<= eps 0.00015167359697961248)
     (*
      eps
      (-
       (* eps (+ (* eps (* (sin x) 0.16666666666666666)) (* (cos x) -0.5)))
       (sin x)))
     (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0001675426203356274) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else if (eps <= 0.00015167359697961248) {
		tmp = eps * ((eps * ((eps * (sin(x) * 0.16666666666666666)) + (cos(x) * -0.5))) - sin(x));
	} else {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.5
Cost32904
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018218815574185185 \lor \neg \left(\varepsilon \leq 0.00015167359697961248\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)\\ \end{array}\]
Alternative 2
Error0.5
Cost26504
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018218815574185185 \lor \neg \left(\varepsilon \leq 0.00015167359697961248\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)\\ \end{array}\]
Alternative 3
Error14.9
Cost13632
\[-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\]
Alternative 4
Error14.6
Cost14018
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4794567668379724 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.10441087167263563:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array}\]
Alternative 5
Error20.8
Cost13512
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.767339696354591 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 4.073923632484895 \cdot 10^{-09}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array}\]
Alternative 6
Error20.7
Cost13634
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1735105876753362 \cdot 10^{-05}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 2.1045680883946606 \cdot 10^{-08}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array}\]
Alternative 7
Error21.0
Cost6984
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.674656834469873 \cdot 10^{-06} \lor \neg \left(\varepsilon \leq 2.1045680883946606 \cdot 10^{-08}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array}\]
Alternative 8
Error39.5
Cost6592
\[\cos \varepsilon + -1\]
Alternative 9
Error52.3
Cost706
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8977243412453404 \cdot 10^{-69}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 6.397017030345022 \cdot 10^{-67}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 10
Error56.2
Cost64
\[0\]
Alternative 11
Error61.0
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.6754262033562739e-4

    1. Initial program 29.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum_binary64_8940.9

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

    4. Applied associate--l-_binary64_6980.9

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

    5. Simplified0.9

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

    6. Simplified0.9

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

    if -1.6754262033562739e-4 < eps < 1.51673596979612482e-4

    1. Initial program 50.1

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) - \left(\sin x \cdot \varepsilon + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)}\]

    if 1.51673596979612482e-4 < eps

    1. Initial program 29.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum_binary64_8940.8

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

    4. Simplified0.8

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001675426203356274:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015167359697961248:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right) + \cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))