Average Error: 36.9 → 0.4
Time: 9.0s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0007736904031470145 \lor \neg \left(\varepsilon \leq 0.0007069371478589407\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \varepsilon \cdot \cos x\right) - \left(0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)\right)\\ \end{array} \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0007736904031470145 \lor \neg \left(\varepsilon \leq 0.0007069371478589407\right):\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{else}:\\
\;\;\;\;\left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \varepsilon \cdot \cos x\right) - \left(0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0007736904031470145) (not (<= eps 0.0007069371478589407)))
   (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))
   (-
    (+ (* 0.041666666666666664 (* (sin x) (pow eps 4.0))) (* eps (cos x)))
    (+
     (* 0.16666666666666666 (* (cos x) (pow eps 3.0)))
     (* 0.5 (* (sin x) (pow eps 2.0)))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0007736904031470145) || !(eps <= 0.0007069371478589407)) {
		tmp = ((sin(x) * cos(eps)) + (cos(x) * sin(eps))) - sin(x);
	} else {
		tmp = ((0.041666666666666664 * (sin(x) * pow(eps, 4.0))) + (eps * cos(x))) - ((0.16666666666666666 * (cos(x) * pow(eps, 3.0))) + (0.5 * (sin(x) * pow(eps, 2.0))));
	}
	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

Target

Original36.9
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if eps < -7.7369040314701452e-4 or 7.0693714785894073e-4 < eps

    1. Initial program 29.6

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

    2. Applied sin-sum_binary640.6

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

    if -7.7369040314701452e-4 < eps < 7.0693714785894073e-4

    1. Initial program 44.3

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

    2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \varepsilon \cdot \cos x\right) - \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0007736904031470145 \lor \neg \left(\varepsilon \leq 0.0007069371478589407\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \varepsilon \cdot \cos x\right) - \left(0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 1991292424 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))