Average Error: 36.9 → 0.4
Time: 11.2s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\begin{array}{l} t_0 := \sin x \cdot \cos \varepsilon\\ t_1 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.7466463302217585 \cdot 10^{-7}:\\ \;\;\;\;\left(t_0 + t_1\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.2208837136567138 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(t_1 - \sin x\right)\\ \end{array} \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
t_0 := \sin x \cdot \cos \varepsilon\\
t_1 := \cos x \cdot \sin \varepsilon\\
\mathbf{if}\;\varepsilon \leq -1.7466463302217585 \cdot 10^{-7}:\\
\;\;\;\;\left(t_0 + t_1\right) - \sin x\\

\mathbf{elif}\;\varepsilon \leq 1.2208837136567138 \cdot 10^{-8}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 + \left(t_1 - \sin x\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (cos eps))) (t_1 (* (cos x) (sin eps))))
   (if (<= eps -1.7466463302217585e-7)
     (- (+ t_0 t_1) (sin x))
     (if (<= eps 1.2208837136567138e-8)
       (* 2.0 (* (sin (/ eps 2.0)) (cos (/ (fma x 2.0 eps) 2.0))))
       (+ t_0 (- t_1 (sin x)))))))
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);
	double t_1 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -1.7466463302217585e-7) {
		tmp = (t_0 + t_1) - sin(x);
	} else if (eps <= 1.2208837136567138e-8) {
		tmp = 2.0 * (sin(eps / 2.0) * cos(fma(x, 2.0, eps) / 2.0));
	} else {
		tmp = t_0 + (t_1 - sin(x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
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 3 regimes

  2. if eps < -1.74664633022175853e-7

    1. Initial program 30.1

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

    2. Applied sin-sum_binary640.5

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

    if -1.74664633022175853e-7 < eps < 1.22088371365671383e-8

    1. Initial program 44.3

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

    2. Applied diff-sin_binary6444.3

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

    3. Simplified0.3

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

    4. Taylor expanded in x around 0 0.3

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

    5. Simplified0.3

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(x, 2, \varepsilon\right)}}{2}\right)\right) \]

    if 1.22088371365671383e-8 < eps

    1. Initial program 29.6

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

    2. Applied sin-sum_binary640.5

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

    3. Applied associate--l+_binary640.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7466463302217585 \cdot 10^{-7}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.2208837136567138 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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)))