Average Error: 7.9 → 0.6
Time: 5.9s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -1.2661495055598136 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 9.070213117930512 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x \cdot 2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\
\mathbf{if}\;t_0 \leq -1.2661495055598136 \cdot 10^{-87}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{elif}\;t_0 \leq 9.070213117930512 \cdot 10^{-10}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x \cdot 2}}{z}\\

\mathbf{else}:\\
\;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 -1.2661495055598136e-87)
     (/ (/ (* (cosh x) y) z) x)
     (if (<= t_0 9.070213117930512e-10)
       (/ (/ (fma y (exp x) (/ y (exp x))) (* x 2.0)) z)
       (* (cosh x) (/ (/ y z) x))))))
double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -1.2661495055598136e-87) {
		tmp = ((cosh(x) * y) / z) / x;
	} else if (t_0 <= 9.070213117930512e-10) {
		tmp = (fma(y, exp(x), (y / exp(x))) / (x * 2.0)) / z;
	} else {
		tmp = cosh(x) * ((y / z) / x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.9
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -1.2661495055598136e-87

    1. Initial program 11.1

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

    2. Applied associate-*r/_binary6411.0

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

    3. Applied associate-/l/_binary649.6

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

    4. Applied associate-/r*_binary641.1

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

    if -1.2661495055598136e-87 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.0702131179305117e-10

    1. Initial program 0.2

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

    2. Applied cosh-def_binary640.2

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

    3. Applied frac-times_binary640.2

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

    4. Simplified0.2

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

    5. Simplified0.2

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

    if 9.0702131179305117e-10 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

    1. Initial program 12.6

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

    2. Applied associate-*r/_binary6412.6

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

    3. Applied associate-/l/_binary6411.6

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

    4. Applied associate-/r*_binary640.3

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

    5. Applied *-un-lft-identity_binary640.3

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

    6. Applied *-un-lft-identity_binary640.3

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

    7. Applied times-frac_binary640.3

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

    8. Applied times-frac_binary640.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -1.2661495055598136 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 9.070213117930512 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x \cdot 2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))