Average Error: 36.9 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}, \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}, \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), (1.0 / (fma(cos(eps), (cos(eps) + 1.0), 1.0) / (pow(cos(eps), 3.0) - 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

Original36.9
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 36.9

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

  3. Applied sin-sum21.7

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

  4. Taylor expanded around inf 21.7

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

  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. Using strategy rm

  11. Applied clear-num0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +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)))