Average Error: 36.8 → 0.4
Time: 5.5s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1685964086771769 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 5.519874034867513 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \left(\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon\right)}{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1.1685964086771769 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 5.519874034867513 \cdot 10^{-38}\right):\\
\;\;\;\;\frac{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \left(\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon\right)}{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\

\end{array}
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1685964086771769e-08) (not (<= eps 5.519874034867513e-38)))
   (/
    (*
     (+ (* (cos x) (sin eps)) (* (sin x) (+ (cos eps) -1.0)))
     (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps))))
    (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps))))
   (* 2.0 (* (sin (/ eps 2.0)) (cos (/ (+ x (+ eps x)) 2.0))))))
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	double tmp;
	if (((eps <= -1.1685964086771769e-08) || !(eps <= 5.519874034867513e-38))) {
		tmp = (((double) (((double) (((double) (((double) cos(x)) * ((double) sin(eps)))) + ((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + -1.0)))))) * ((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps)))))))) / ((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps)))))));
	} else {
		tmp = ((double) (2.0 * ((double) (((double) sin((eps / 2.0))) * ((double) cos((((double) (x + ((double) (eps + x)))) / 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

Original36.8
Target14.9
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 < -1.16859640867717687e-8 or 5.519874034867513e-38 < eps

    1. Initial program 29.5

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

    3. Applied sin-sum_binary641.7

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

    4. Applied associate--l+_binary641.7

      \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    5. Using strategy rm

    6. Applied flip-+_binary641.9

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

    7. Simplified0.8

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

    8. Simplified0.6

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

    if -1.16859640867717687e-8 < eps < 5.519874034867513e-38

    1. Initial program 45.3

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

    3. Applied diff-sin_binary6445.3

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

    4. Simplified0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1685964086771769 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 5.519874034867513 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \left(\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon\right)}{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))