Average Error: 37.0 → 0.3
Time: 6.6s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023068637899912437 \lor \neg \left(\varepsilon \leq 1.2826844230892183 \cdot 10^{-5}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right) + \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.00023068637899912437 \lor \neg \left(\varepsilon \leq 1.2826844230892183 \cdot 10^{-5}\right):\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\

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


\end{array}

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00023068637899912437) (not (<= eps 1.2826844230892183e-5)))
   (+ (* (sin x) (cos eps)) (- (* (sin eps) (cos x)) (sin x)))
   (+
    (* eps (* -0.5 (* eps (sin x))))
    (* (cos x) (+ eps (* (pow eps 3.0) -0.16666666666666666))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00023068637899912437) || !(eps <= 1.2826844230892183e-5)) {
		tmp = (sin(x) * cos(eps)) + ((sin(eps) * cos(x)) - sin(x));
	} else {
		tmp = (eps * (-0.5 * (eps * sin(x)))) + (cos(x) * (eps + (pow(eps, 3.0) * -0.16666666666666666)));
	}
	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
Target15.1
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 < -2.3068637899912437e-4 or 1.2826844230892183e-5 < eps

    1. Initial program 29.7

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

    3. Applied sin-sum_binary640.4

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

    4. Applied associate--l+_binary640.5

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

    5. Simplified0.5

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

    if -2.3068637899912437e-4 < eps < 1.2826844230892183e-5

    1. Initial program 44.8

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023068637899912437 \lor \neg \left(\varepsilon \leq 1.2826844230892183 \cdot 10^{-5}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right) + \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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)))