Average Error: 37.1 → 0.3
Time: 9.5s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\begin{array}{l} t_0 := \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0006848596310522501:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0007902110569607544:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \cos x, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right), \left(\cos x \cdot {\varepsilon}^{3}\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
t_0 := \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\
\mathbf{if}\;\varepsilon \leq -0.0006848596310522501:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\varepsilon \leq 0.0007902110569607544:\\
\;\;\;\;\mathsf{fma}\left(\varepsilon, \cos x, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right), \left(\cos x \cdot {\varepsilon}^{3}\right) \cdot -0.16666666666666666\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))))
   (if (<= eps -0.0006848596310522501)
     t_0
     (if (<= eps 0.0007902110569607544)
       (fma
        eps
        (cos x)
        (fma
         (sin x)
         (fma 0.041666666666666664 (pow eps 4.0) (* (* eps eps) -0.5))
         (* (* (cos x) (pow eps 3.0)) -0.16666666666666666)))
       t_0))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	double t_0 = ((sin(x) * cos(eps)) + (cos(x) * sin(eps))) - sin(x);
	double tmp;
	if (eps <= -0.0006848596310522501) {
		tmp = t_0;
	} else if (eps <= 0.0007902110569607544) {
		tmp = fma(eps, cos(x), fma(sin(x), fma(0.041666666666666664, pow(eps, 4.0), ((eps * eps) * -0.5)), ((cos(x) * pow(eps, 3.0)) * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if eps < -6.8485963105225007e-4 or 7.90211056960754446e-4 < eps

    1. Initial program 29.6

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

    2. Applied sin-sum_binary640.4

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

    if -6.8485963105225007e-4 < eps < 7.90211056960754446e-4

    1. Initial program 44.8

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

    2. Taylor expanded in eps around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0006848596310522501:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0007902110569607544:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \cos x, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right), \left(\cos x \cdot {\varepsilon}^{3}\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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)))