Average Error: 37.0 → 0.4
Time: 24.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)
double f(double x, double eps) {
        double r3058114 = x;
        double r3058115 = eps;
        double r3058116 = r3058114 + r3058115;
        double r3058117 = sin(r3058116);
        double r3058118 = sin(r3058114);
        double r3058119 = r3058117 - r3058118;
        return r3058119;
}
↓
double f(double x, double eps) {
        double r3058120 = 2.0;
        double r3058121 = x;
        double r3058122 = cos(r3058121);
        double r3058123 = eps;
        double r3058124 = 0.5;
        double r3058125 = r3058123 * r3058124;
        double r3058126 = cos(r3058125);
        double r3058127 = r3058122 * r3058126;
        double r3058128 = sin(r3058121);
        double r3058129 = sin(r3058125);
        double r3058130 = r3058128 * r3058129;
        double r3058131 = r3058130 * r3058130;
        double r3058132 = r3058130 * r3058131;
        double r3058133 = cbrt(r3058132);
        double r3058134 = r3058127 - r3058133;
        double r3058135 = r3058134 * r3058129;
        double r3058136 = r3058120 * r3058135;
        return r3058136;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied diff-sin37.3

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

  4. Simplified15.0

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

  5. Taylor expanded around -inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  9. Applied cos-sum0.3

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

  11. Applied add-cbrt-cube0.4

    \[\leadsto 2 \cdot \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  12. Final simplification0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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