Average Error: 0.1 → 0.2
Time: 15.4s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\sqrt{e^{x} + e^{-x}} \cdot \left(\sqrt{\cosh x} \cdot \frac{\sin y}{y}\right)}{\sqrt{2}}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\sqrt{e^{x} + e^{-x}} \cdot \left(\sqrt{\cosh x} \cdot \frac{\sin y}{y}\right)}{\sqrt{2}}
double f(double x, double y) {
        double r454701 = x;
        double r454702 = cosh(r454701);
        double r454703 = y;
        double r454704 = sin(r454703);
        double r454705 = r454704 / r454703;
        double r454706 = r454702 * r454705;
        return r454706;
}
↓
double f(double x, double y) {
        double r454707 = x;
        double r454708 = exp(r454707);
        double r454709 = -r454707;
        double r454710 = exp(r454709);
        double r454711 = r454708 + r454710;
        double r454712 = sqrt(r454711);
        double r454713 = cosh(r454707);
        double r454714 = sqrt(r454713);
        double r454715 = y;
        double r454716 = sin(r454715);
        double r454717 = r454716 / r454715;
        double r454718 = r454714 * r454717;
        double r454719 = r454712 * r454718;
        double r454720 = 2.0;
        double r454721 = sqrt(r454720);
        double r454722 = r454719 / r454721;
        return r454722;
}

Error

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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. Using strategy rm

  3. Applied clear-num0.2

    \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.2

    \[\leadsto \color{blue}{\left(\sqrt{\cosh x} \cdot \sqrt{\cosh x}\right)} \cdot \frac{1}{\frac{y}{\sin y}}\]

  6. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{1}{\frac{y}{\sin y}}\right)}\]

  7. Simplified0.2

    \[\leadsto \sqrt{\cosh x} \cdot \color{blue}{\left(\sqrt{\cosh x} \cdot \frac{\sin y}{y}\right)}\]
  8. Using strategy rm

  9. Applied cosh-def0.2

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

  10. Applied sqrt-div0.2

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

  11. Applied associate-*l/0.2

    \[\leadsto \sqrt{\cosh x} \cdot \color{blue}{\frac{\sqrt{e^{x} + e^{-x}} \cdot \frac{\sin y}{y}}{\sqrt{2}}}\]

  12. Applied associate-*r/0.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))