Average Error: 36.9 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}, \cos x \cdot \sin \varepsilon\right)
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	return ((double) fma(((double) sin(x)), ((double) (((double) (((double) pow(((double) cos(eps)), 3.0)) - 1.0)) / ((double) fma(((double) cos(eps)), ((double) (((double) cos(eps)) + 1.0)), 1.0)))), ((double) (((double) cos(x)) * ((double) 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
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.5

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

  4. Taylor expanded around inf 21.5

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

  5. Simplified0.4

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

  7. Applied flip3--0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020120 +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)))