Average Error: 7.7 → 0.4
Time: 28.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\
\;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\

\end{array}
double f(double x, double y, double z) {
        double r330205 = x;
        double r330206 = cosh(r330205);
        double r330207 = y;
        double r330208 = r330207 / r330205;
        double r330209 = r330206 * r330208;
        double r330210 = z;
        double r330211 = r330209 / r330210;
        return r330211;
}
↓
double f(double x, double y, double z) {
        double r330212 = z;
        double r330213 = -7.575455656273212e+37;
        bool r330214 = r330212 <= r330213;
        double r330215 = 189132.75325345353;
        bool r330216 = r330212 <= r330215;
        double r330217 = !r330216;
        bool r330218 = r330214 || r330217;
        double r330219 = x;
        double r330220 = exp(r330219);
        double r330221 = 3.0;
        double r330222 = pow(r330220, r330221);
        double r330223 = -r330219;
        double r330224 = exp(r330223);
        double r330225 = pow(r330224, r330221);
        double r330226 = r330222 + r330225;
        double r330227 = -2.0;
        double r330228 = r330227 * r330219;
        double r330229 = expm1(r330228);
        double r330230 = fma(r330220, r330220, r330229);
        double r330231 = r330226 / r330230;
        double r330232 = y;
        double r330233 = r330231 * r330232;
        double r330234 = 2.0;
        double r330235 = r330234 * r330219;
        double r330236 = r330212 * r330235;
        double r330237 = r330233 / r330236;
        double r330238 = r330220 + r330224;
        double r330239 = r330238 * r330232;
        double r330240 = r330239 / r330212;
        double r330241 = r330240 / r330235;
        double r330242 = r330218 ? r330237 : r330241;
        return r330242;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.575455656273212e+37 or 189132.75325345353 < z

    1. Initial program 12.5

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied cosh-def12.5

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

    4. Applied frac-times12.5

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

    5. Applied associate-/l/0.3

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

    7. Applied flip3-+0.5

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

    8. Simplified0.5

      \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)}} \cdot y}{z \cdot \left(2 \cdot x\right)}\]

    if -7.575455656273212e+37 < z < 189132.75325345353

    1. Initial program 0.4

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied cosh-def0.4

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

    4. Applied frac-times0.4

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

    5. Applied associate-/l/17.4

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

    7. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))