Average Error: 7.7 → 0.3
Time: 4.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.25394936730307222 \cdot 10^{-15}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 1.617536403644778 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.25394936730307222 \cdot 10^{-15}:\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{elif}\;y \le 1.617536403644778 \cdot 10^{-11}:\\
\;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\

\end{array}
double f(double x, double y, double z) {
        double r487544 = x;
        double r487545 = cosh(r487544);
        double r487546 = y;
        double r487547 = r487546 / r487544;
        double r487548 = r487545 * r487547;
        double r487549 = z;
        double r487550 = r487548 / r487549;
        return r487550;
}
↓
double f(double x, double y, double z) {
        double r487551 = y;
        double r487552 = -6.253949367303072e-15;
        bool r487553 = r487551 <= r487552;
        double r487554 = x;
        double r487555 = cosh(r487554);
        double r487556 = z;
        double r487557 = r487554 * r487556;
        double r487558 = r487551 / r487557;
        double r487559 = r487555 * r487558;
        double r487560 = 1.6175364036447777e-11;
        bool r487561 = r487551 <= r487560;
        double r487562 = sqrt(r487555);
        double r487563 = r487551 / r487554;
        double r487564 = r487562 * r487563;
        double r487565 = r487562 * r487564;
        double r487566 = r487565 / r487556;
        double r487567 = 0.5;
        double r487568 = -1.0;
        double r487569 = r487568 * r487554;
        double r487570 = exp(r487569);
        double r487571 = exp(r487554);
        double r487572 = r487570 + r487571;
        double r487573 = r487567 * r487572;
        double r487574 = r487556 * r487554;
        double r487575 = r487574 / r487551;
        double r487576 = r487573 / r487575;
        double r487577 = r487561 ? r487566 : r487576;
        double r487578 = r487553 ? r487559 : r487577;
        return r487578;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.7
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -6.253949367303072e-15

    1. Initial program 19.1

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

    3. Applied *-un-lft-identity19.1

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac19.1

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

    5. Simplified19.1

      \[\leadsto \color{blue}{\cosh x} \cdot \frac{\frac{y}{x}}{z}\]

    6. Simplified0.4

      \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}}\]

    if -6.253949367303072e-15 < y < 1.6175364036447777e-11

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied associate-*l*0.3

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

    if 1.6175364036447777e-11 < y

    1. Initial program 21.0

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.25394936730307222 \cdot 10^{-15}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 1.617536403644778 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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