Average Error: 37.2 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{\sin \varepsilon \cdot \sin \varepsilon}{-\left(\cos \varepsilon + 1\right)}, \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{\sin \varepsilon \cdot \sin \varepsilon}{-\left(\cos \varepsilon + 1\right)}, \cos x \cdot \sin \varepsilon\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return fma(sin(x), ((sin(eps) * sin(eps)) / -(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

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

Derivation

  1. Initial program 37.2

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Applied associate--l+22.0

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

  5. Taylor expanded around inf 22.0

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

  6. Simplified0.4

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

  8. Applied flip--0.5

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

  9. Simplified0.4

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

  11. Applied frac-2neg0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{\sin \varepsilon \cdot \sin \varepsilon}{-\left(\cos \varepsilon + 1\right)}, \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020103 +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)))