Average Error: 2.7 → 0.4
Time: 5.8s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2.8388904751584647 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{elif}\;z \leq 2.0504899839305107 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
t_0 := \frac{\sin y}{y}\\
\mathbf{if}\;z \leq -2.8388904751584647 \cdot 10^{-127}:\\
\;\;\;\;\frac{x \cdot t_0}{z}\\

\mathbf{elif}\;z \leq 2.0504899839305107 \cdot 10^{-17}:\\
\;\;\;\;\frac{x}{\frac{z}{t_0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\


\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 (<= z -2.8388904751584647e-127)
     (/ (* x t_0) z)
     (if (<= z 2.0504899839305107e-17)
       (/ x (/ z t_0))
       (/ (/ x z) (/ y (sin y)))))))
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 tmp;
	if (z <= -2.8388904751584647e-127) {
		tmp = (x * t_0) / z;
	} else if (z <= 2.0504899839305107e-17) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) / (y / sin(y));
	}
	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

Original2.7
Target0.3
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.83889047515846475e-127

    1. Initial program 0.9

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

    if -2.83889047515846475e-127 < z < 2.0504899839305107e-17

    1. Initial program 6.9

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

    2. Applied associate-/l*_binary640.2

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

    if 2.0504899839305107e-17 < z

    1. Initial program 0.1

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

    2. Applied associate-/l*_binary645.1

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

    3. Applied div-inv_binary645.1

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

    4. Applied associate-/r*_binary640.1

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

    5. Taylor expanded in y around inf 0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8388904751584647 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;z \leq 2.0504899839305107 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))