Average Error: 37.2 → 0.3
Time: 15.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2
double f(double x, double eps) {
        double r3226192 = x;
        double r3226193 = eps;
        double r3226194 = r3226192 + r3226193;
        double r3226195 = sin(r3226194);
        double r3226196 = sin(r3226192);
        double r3226197 = r3226195 - r3226196;
        return r3226197;
}
↓
double f(double x, double eps) {
        double r3226198 = 0.5;
        double r3226199 = eps;
        double r3226200 = r3226198 * r3226199;
        double r3226201 = cos(r3226200);
        double r3226202 = x;
        double r3226203 = cos(r3226202);
        double r3226204 = r3226201 * r3226203;
        double r3226205 = sin(r3226200);
        double r3226206 = sin(r3226202);
        double r3226207 = r3226205 * r3226206;
        double r3226208 = r3226204 - r3226207;
        double r3226209 = r3226208 * r3226205;
        double r3226210 = 2.0;
        double r3226211 = r3226209 * r3226210;
        return r3226211;
}

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
Target14.9
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied diff-sin37.5

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified14.9

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

  5. Taylor expanded around -inf 14.9

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)}\right)\]

  6. Simplified14.9

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\]
  7. Using strategy rm

  8. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\right)\]

  9. Taylor expanded around -inf 0.3

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  10. Final simplification0.3

    \[\leadsto \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))