Average Error: 36.6 → 0.4
Time: 8.3s
Precision: binary64
Cost: 26176
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+ (* (sin eps) (cos x)) (* (sin x) (+ (cos eps) -1.0))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return (sin(eps) * cos(x)) + (sin(x) * (cos(eps) + -1.0));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Alternatives

Alternative 1
Error14.9
Cost13632
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\]
Alternative 2
Error14.3
Cost13704
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4794567668379724 \cdot 10^{-05} \lor \neg \left(\varepsilon \leq 2.1045680883946606 \cdot 10^{-08}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5\right)\right)\\ \end{array}\]
Alternative 3
Error14.4
Cost13320
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1182334377507595 \cdot 10^{-05} \lor \neg \left(\varepsilon \leq 2.1045680883946606 \cdot 10^{-08}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 4
Error14.8
Cost6920
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1182334377507595 \cdot 10^{-05} \lor \neg \left(\varepsilon \leq 2.1045680883946606 \cdot 10^{-08}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 5
Error28.8
Cost6464
\[\sin \varepsilon\]
Alternative 6
Error59.3
Cost64
\[-1\]
Alternative 7
Error59.3
Cost64
\[1\]

Error

Derivation

  1. Initial program 36.6

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

  3. Applied sin-sum_binary64_157521.8

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

  5. Applied *-un-lft-identity_binary64_144221.8

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

  6. Applied *-un-lft-identity_binary64_144221.8

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

  7. Applied distribute-lft-out--_binary64_139421.8

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

    \[\leadsto \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\]

Reproduce

herbie shell --seed 2021044 
(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)))