Average Error: 37.1 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r118941 = x;
        double r118942 = eps;
        double r118943 = r118941 + r118942;
        double r118944 = sin(r118943);
        double r118945 = sin(r118941);
        double r118946 = r118944 - r118945;
        return r118946;
}
↓
double f(double x, double eps) {
        double r118947 = x;
        double r118948 = sin(r118947);
        double r118949 = eps;
        double r118950 = cos(r118949);
        double r118951 = r118948 * r118950;
        double r118952 = -r118948;
        double r118953 = r118951 + r118952;
        double r118954 = cos(r118947);
        double r118955 = sin(r118949);
        double r118956 = r118954 * r118955;
        double r118957 = r118953 + r118956;
        return r118957;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum21.9

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

  4. Taylor expanded around inf 21.9

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

  5. Simplified0.4

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

  7. Applied sub-neg0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020046 
(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)))