Average Error: 37.1 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \log \left(e^{\cos \varepsilon - 1}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \log \left(e^{\cos \varepsilon - 1}\right) + \cos x \cdot \sin \varepsilon
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return ((sin(x) * log(exp((cos(eps) - 1.0)))) + (cos(x) * sin(eps)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum22.5

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

  5. Applied *-un-lft-identity22.5

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

  6. Applied *-un-lft-identity22.5

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

  7. Applied distribute-lft-out--22.5

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

  8. Simplified0.4

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

  10. Applied add-log-exp0.4

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

  11. Applied add-log-exp0.4

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

  12. Applied diff-log0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020066 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))