Average Error: 2.8 → 0.2
Time: 4.9s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\begin{array}{l} t_1 := x \cdot t_0\\ t_1 \leq -8.162421889339563 \cdot 10^{-272} \lor \neg \left(t_1 \leq 5.206480742556539 \cdot 10^{-212}\right) \end{array}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
t_0 := \frac{\sin y}{y}\\
\mathbf{if}\;\begin{array}{l}
t_1 := x \cdot t_0\\
t_1 \leq -8.162421889339563 \cdot 10^{-272} \lor \neg \left(t_1 \leq 5.206480742556539 \cdot 10^{-212}\right)
\end{array}:\\
\;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{t_0}{z}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (let* ((t_1 (* x t_0)))
         (or (<= t_1 -8.162421889339563e-272)
             (not (<= t_1 5.206480742556539e-212))))
     (/ (* x (/ 1.0 (/ y (sin y)))) z)
     (* x (/ t_0 z)))))
double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -8.162421889339563e-272) || !(t_1 <= 5.206480742556539e-212)) {
		tmp = (x * (1.0 / (y / sin(y)))) / z;
	} else {
		tmp = x * (t_0 / z);
	}
	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

Original2.8
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -8.1624218893395627e-272 or 5.20648074255653865e-212 < (*.f64 x (/.f64 (sin.f64 y) y))

    1. Initial program 0.2

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

    2. Applied clear-num_binary640.2

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

    if -8.1624218893395627e-272 < (*.f64 x (/.f64 (sin.f64 y) y)) < 5.20648074255653865e-212

    1. Initial program 10.2

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

    2. Applied *-un-lft-identity_binary6410.2

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

    3. Applied times-frac_binary640.3

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\frac{\sin y}{y}}{z}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -8.162421889339563 \cdot 10^{-272} \lor \neg \left(x \cdot \frac{\sin y}{y} \leq 5.206480742556539 \cdot 10^{-212}\right):\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))