Average Error: 37.0 → 0.4
Time: 5.6s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1949421538796511 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 7.799591496688593 \cdot 10^{-09}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1.1949421538796511 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 7.799591496688593 \cdot 10^{-09}\right):\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1949421538796511e-08) (not (<= eps 7.799591496688593e-09)))
   (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))
   (* (* 2.0 (sin (/ eps 2.0))) (cos (/ (+ x (+ eps x)) 2.0)))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1949421538796511e-08) || !(eps <= 7.799591496688593e-09)) {
		tmp = ((sin(x) * cos(eps)) + (cos(x) * sin(eps))) - sin(x);
	} else {
		tmp = (2.0 * sin(eps / 2.0)) * cos((x + (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

Original37.0
Target15.0
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 < -1.1949421538796511e-8 or 7.79959149668859336e-9 < eps

    1. Initial program 29.7

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

    3. Applied sin-sum_binary64_18920.6

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

    if -1.1949421538796511e-8 < eps < 7.79959149668859336e-9

    1. Initial program 44.7

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

    3. Applied diff-sin_binary64_190944.7

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

    4. Simplified0.3

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

    6. Applied associate-*r*_binary64_17040.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1949421538796511 \cdot 10^{-08} \lor \neg \left(\varepsilon \leq 7.799591496688593 \cdot 10^{-09}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))