Average Error: 36.8 → 0.3
Time: 53.2s
Precision: binary64
Cost: 33218
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0009986885707327965:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0007460669567238816:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.020833333333333332\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0009986885707327965:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \leq 0.0007460669567238816:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.020833333333333332\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0009986885707327965)
   (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))
   (if (<= eps 0.0007460669567238816)
     (*
      2.0
      (*
       (sin (/ eps 2.0))
       (+
        (cos x)
        (*
         eps
         (-
          (* (sin x) (+ -0.5 (* (* eps eps) 0.020833333333333332)))
          (* eps (* (cos x) 0.125)))))))
     (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0009986885707327965) {
		tmp = ((sin(x) * cos(eps)) + (cos(x) * sin(eps))) - sin(x);
	} else if (eps <= 0.0007460669567238816) {
		tmp = 2.0 * (sin(eps / 2.0) * (cos(x) + (eps * ((sin(x) * (-0.5 + ((eps * eps) * 0.020833333333333332))) - (eps * (cos(x) * 0.125))))));
	} else {
		tmp = (sin(x) * cos(eps)) + ((cos(x) * sin(eps)) - sin(x));
	}
	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
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Alternatives

Alternative 1
Error0.3
Cost32904
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0008790196068280468 \lor \neg \left(\varepsilon \leq 0.000701317829132117\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.020833333333333332\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)\right)\\ \end{array}\]
Alternative 2
Error15.0
Cost13632
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\]
Alternative 3
Error14.9
Cost13504
\[2 \cdot \left(\cos \left(x + \varepsilon \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\]
Alternative 4
Error14.7
Cost39688
\[\begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.00044915867917528406 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 8.771108688767795 \cdot 10^{-78}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]
Alternative 5
Error14.6
Cost13320
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025328896187052515 \lor \neg \left(\varepsilon \leq 0.00011414825082033838\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 6
Error15.0
Cost6920
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025328896187052515 \lor \neg \left(\varepsilon \leq 0.00016216259863038255\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 7
Error28.6
Cost6464
\[\sin \varepsilon\]
Alternative 8
Error42.7
Cost706
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.006750103867893:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 9.458511693510916:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 9
Error45.2
Cost64
\[\varepsilon\]

Error

Derivation

  1. Split input into 3 regimes

  2. if eps < -9.9868857073279649e-4

    1. Initial program 30.2

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

    3. Applied pow1_binary64_184430.2

      \[\leadsto \color{blue}{{\sin \left(x + \varepsilon\right)}^{1}} - \sin x\]
    4. Using strategy rm

    5. Applied sin-sum_binary64_19160.4

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

    6. Simplified0.4

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

    if -9.9868857073279649e-4 < eps < 7.4606695672388163e-4

    1. Initial program 44.0

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

    3. Applied diff-sin_binary64_193344.0

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

    4. Simplified0.5

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

    5. Taylor expanded around 0 0.1

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(\cos x + 0.020833333333333332 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(0.5 \cdot \left(\sin x \cdot \varepsilon\right) + 0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\]

    6. Simplified0.1

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(0.020833333333333332 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)}\right)\]

    7. Simplified0.1

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.020833333333333332\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)\right)}\]

    if 7.4606695672388163e-4 < eps

    1. Initial program 29.2

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

    3. Applied sin-sum_binary64_19160.4

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

    4. Applied associate--l+_binary64_17200.4

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

    5. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0009986885707327965:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0007460669567238816:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot \left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.020833333333333332\right) - \varepsilon \cdot \left(\cos x \cdot 0.125\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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)))