Average Error: 36.7 → 0.4
Time: 5.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin \varepsilon \cdot \cos x + \mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return ((sin(eps) * cos(x)) + fma(cos(eps), sin(x), -sin(x)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.7

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

  3. Applied +-commutative36.7

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

  4. Applied sin-sum21.7

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

  5. Applied associate--l+0.4

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

  7. Applied fma-neg0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020071 +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)))