Average Error: 2.8 → 0.3
Time: 5.1s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1.0897289187131854 \cdot 10^{-154}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{y}{\sin y}\\ \mathbf{if}\;t_0 \leq 3.588673077855241 \cdot 10^{-233}:\\ \;\;\;\;\frac{\frac{x}{z}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{z}\\ \end{array}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
t_0 := x \cdot \frac{\sin y}{y}\\
\mathbf{if}\;t_0 \leq -1.0897289187131854 \cdot 10^{-154}:\\
\;\;\;\;\frac{t_0}{z}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{y}{\sin y}\\
\mathbf{if}\;t_0 \leq 3.588673077855241 \cdot 10^{-233}:\\
\;\;\;\;\frac{\frac{x}{z}}{t_1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{t_1}}{z}\\


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (sin y) y))))
   (if (<= t_0 -1.0897289187131854e-154)
     (/ t_0 z)
     (let* ((t_1 (/ y (sin y))))
       (if (<= t_0 3.588673077855241e-233) (/ (/ x z) t_1) (/ (/ x t_1) 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 = x * (sin(y) / y);
	double tmp;
	if (t_0 <= -1.0897289187131854e-154) {
		tmp = t_0 / z;
	} else {
		double t_1 = y / sin(y);
		double tmp_1;
		if (t_0 <= 3.588673077855241e-233) {
			tmp_1 = (x / z) / t_1;
		} else {
			tmp_1 = (x / t_1) / z;
		}
		tmp = tmp_1;
	}
	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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -1.08972891871318537e-154

    1. Initial program 0.2

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

    if -1.08972891871318537e-154 < (*.f64 x (/.f64 (sin.f64 y) y)) < 3.588673077855241e-233

    1. Initial program 7.8

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

    2. Applied clear-num_binary647.8

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

    3. Applied un-div-inv_binary647.8

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

    4. Applied associate-/l/_binary640.5

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

    5. Applied associate-/r*_binary640.4

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

    if 3.588673077855241e-233 < (*.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

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

    3. Applied un-div-inv_binary640.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1.0897289187131854 \cdot 10^{-154}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 3.588673077855241 \cdot 10^{-233}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))