Average Error: 36.9 → 0.8
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return fma(sin(x), log(exp((cos(eps) - 1.0))), ((cbrt(cos(x)) * cbrt(cos(x))) * (cbrt(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

Original36.9
Target14.8
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum22.0

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

  5. Applied *-un-lft-identity22.0

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

  6. Applied *-un-lft-identity22.0

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

  7. Applied distribute-lft-out--22.0

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

  8. Simplified0.4

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

  10. Applied add-cube-cbrt0.7

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \sin \varepsilon\right)\]

  11. Applied associate-*l*0.7

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)}\right)\]
  12. Using strategy rm

  13. Applied add-log-exp0.7

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \cos \varepsilon - \color{blue}{\log \left(e^{1}\right)}, \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]

  14. Applied add-log-exp0.8

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \color{blue}{\log \left(e^{\cos \varepsilon}\right)} - \log \left(e^{1}\right), \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]

  15. Applied diff-log0.8

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \color{blue}{\log \left(\frac{e^{\cos \varepsilon}}{e^{1}}\right)}, \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]

  16. Simplified0.8

    \[\leadsto 1 \cdot \mathsf{fma}\left(\sin x, \log \color{blue}{\left(e^{\cos \varepsilon - 1}\right)}, \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]

  17. Final simplification0.8

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(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)))