Average Error: 7.9 → 0.5
Time: 4.1s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} = -inf.0:\\ \;\;\;\;y \cdot \left(\frac{e^{x} + e^{-x}}{x} \cdot \frac{\frac{1}{2}}{z}\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.69207147685910959 \cdot 10^{155}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{\cosh x \cdot y}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.9
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (cosh x) (/ y x)) < -inf.0

    1. Initial program 64.0

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

    if -inf.0 < (* (cosh x) (/ y x)) < 2.69207147685910959e155

    1. Initial program 0.3

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

    if 2.69207147685910959e155 < (* (cosh x) (/ y x))

    1. Initial program 23.2

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

    3. Applied clear-num23.3

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

    4. Simplified1.9

      \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{z}{\cosh x \cdot y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} = -inf.0:\\ \;\;\;\;y \cdot \left(\frac{e^{x} + e^{-x}}{x} \cdot \frac{\frac{1}{2}}{z}\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.69207147685910959 \cdot 10^{155}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{\cosh x \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))