Average Error: 7.9 → 0.3
Time: 3.6s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -3.6887628113558124 \cdot 10^{-38} \lor \neg \left(y \leq 5.543226388400742 \cdot 10^{-34}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{z \cdot 2}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -3.6887628113558124 \cdot 10^{-38} \lor \neg \left(y \leq 5.543226388400742 \cdot 10^{-34}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.6887628113558124e-38) (not (<= y 5.543226388400742e-34)))
   (* (cosh x) (/ y (* x z)))
   (/ (* (+ (exp x) (exp (- x))) (/ y x)) (* z 2.0))))
double code(double x, double y, double z) {
	return (((double) (((double) cosh(x)) * (y / x))) / z);
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (((y <= -3.6887628113558124e-38) || !(y <= 5.543226388400742e-34))) {
		tmp = ((double) (((double) cosh(x)) * (y / ((double) (x * z)))));
	} else {
		tmp = (((double) (((double) (((double) exp(x)) + ((double) exp(((double) -(x)))))) * (y / x))) / ((double) (z * 2.0)));
	}
	return tmp;
}

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

Original7.9
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.6887628113558124e-38 or 5.5432263884007423e-34 < y

    1. Initial program Error: 18.4 bits

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

    2. SimplifiedError: 0.4 bits

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

    if -3.6887628113558124e-38 < y < 5.5432263884007423e-34

    1. Initial program Error: 0.2 bits

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

    3. Applied cosh-defError: 0.2 bits

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

    4. Applied associate-*l/Error: 0.2 bits

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

    5. Applied associate-/l/Error: 0.2 bits

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

  4. Final simplificationError: 0.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6887628113558124 \cdot 10^{-38} \lor \neg \left(y \leq 5.543226388400742 \cdot 10^{-34}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{z \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))