Average Error: 0.1 → 0.2
Time: 27.0s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\left(\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot \frac{1}{2}\right) \cdot \frac{\sin y}{y}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\left(\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot \frac{1}{2}\right) \cdot \frac{\sin y}{y}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}
double f(double x, double y) {
        double r386685 = x;
        double r386686 = cosh(r386685);
        double r386687 = y;
        double r386688 = sin(r386687);
        double r386689 = r386688 / r386687;
        double r386690 = r386686 * r386689;
        return r386690;
}
↓
double f(double x, double y) {
        double r386691 = x;
        double r386692 = exp(r386691);
        double r386693 = 3.0;
        double r386694 = pow(r386692, r386693);
        double r386695 = -r386691;
        double r386696 = exp(r386695);
        double r386697 = pow(r386696, r386693);
        double r386698 = r386694 + r386697;
        double r386699 = 0.5;
        double r386700 = r386698 * r386699;
        double r386701 = y;
        double r386702 = sin(r386701);
        double r386703 = r386702 / r386701;
        double r386704 = r386700 * r386703;
        double r386705 = r386692 * r386692;
        double r386706 = r386696 * r386696;
        double r386707 = r386692 * r386696;
        double r386708 = r386706 - r386707;
        double r386709 = r386705 + r386708;
        double r386710 = r386704 / r386709;
        return r386710;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.1
Target0.1
Herbie0.2
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.1

    \[\cosh x \cdot \frac{\sin y}{y}\]

  2. Taylor expanded around inf 0.1

    \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)} \cdot \frac{\sin y}{y}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}\right)} \cdot \frac{\sin y}{y}\]
  4. Using strategy rm

  5. Applied flip3-+0.2

    \[\leadsto \left(\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}} \cdot \frac{1}{2}\right) \cdot \frac{\sin y}{y}\]

  6. Applied associate-*l/0.2

    \[\leadsto \color{blue}{\frac{\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot \frac{1}{2}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}} \cdot \frac{\sin y}{y}\]

  7. Applied associate-*l/0.2

    \[\leadsto \color{blue}{\frac{\left(\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot \frac{1}{2}\right) \cdot \frac{\sin y}{y}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}\]

  8. Final simplification0.2

    \[\leadsto \frac{\left(\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot \frac{1}{2}\right) \cdot \frac{\sin y}{y}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))