Average Error: 37.3 → 0.4
Time: 1.4min
Precision: binary64
Cost: 52032
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{e^{\log \left(\cos \varepsilon + 1\right)}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{e^{\log \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) (* (sin eps) (sin eps))) (exp (log (+ (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) * (sin(eps) * sin(eps))) / exp(log(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

Original37.3
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Alternatives

Alternative 1
Error0.4
Cost39168
\[\sin \varepsilon \cdot \cos x + \frac{\sin x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\]
Alternative 2
Error0.4
Cost32576
\[\sin \varepsilon \cdot \cos x + \left(\sin x \cdot \cos \varepsilon - \sin x\right)\]
Alternative 3
Error0.4
Cost26176
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\]
Alternative 4
Error14.1
Cost26690
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.14127533727766742:\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.02822313514968131:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \left(0.5 - \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right) - \varepsilon \cdot \left(\sin x \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \varepsilon + \sin x \cdot \cos \varepsilon\right) - \sin x\\ \end{array}\]
Alternative 5
Error14.1
Cost19976
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.005528182842280636 \lor \neg \left(\varepsilon \leq 0.026056827129505016\right):\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot \left(\cos x \cdot \left(0.5 - \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right) - \varepsilon \cdot \left(\sin x \cdot 0.25\right)\right)\right)\\ \end{array}\]
Alternative 6
Error15.1
Cost13504
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)\]
Alternative 7
Error15.0
Cost13576
\[\begin{array}{l} \mathbf{if}\;x \leq -0.002102626412248292 \lor \neg \left(x \leq 0.0009627452477608859\right):\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array}\]
Alternative 8
Error15.1
Cost13320
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -112837442.24806044 \lor \neg \left(\varepsilon \leq 2.3195472044197997 \cdot 10^{-11}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 9
Error15.3
Cost6920
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1115726505891959 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 2.3195472044197997 \cdot 10^{-11}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array}\]
Alternative 10
Error29.0
Cost6464
\[\sin \varepsilon\]
Alternative 11
Error42.9
Cost706
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -764806318.5569421:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 3.337728541767795 \cdot 10^{+16}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 12
Error44.1
Cost385
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1042947682.9242277:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array}\]
Alternative 13
Error45.4
Cost64
\[\varepsilon\]
Alternative 14
Error61.3
Cost64
\[0\]

Error

Time

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum_binary64_157521.9

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

  5. Applied *-un-lft-identity_binary64_144221.9

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

  6. Applied *-un-lft-identity_binary64_144221.9

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

  7. Applied distribute-lft-out--_binary64_139421.9

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

  8. Simplified0.4

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

  10. Applied flip-+_binary64_14160.5

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

  11. Applied associate-*r/_binary64_13840.5

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

  12. Simplified0.4

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

  14. Applied add-exp-log_binary64_14800.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

    \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{e^{\log \left(\cos \varepsilon + 1\right)}}\]

Reproduce

herbie shell --seed 2021040 
(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)))