Average Error: 2.8 → 0.4
Time: 5.0s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.6559076628097083 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{y}{\sin y}\\ \mathbf{if}\;x \leq 1.6643543314697557 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{z}\\ \end{array}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.6559076628097083 \cdot 10^{-102}:\\
\;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{y}{\sin y}\\
\mathbf{if}\;x \leq 1.6643543314697557 \cdot 10^{-7}:\\
\;\;\;\;\frac{x}{z \cdot t_0}\\

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


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6559076628097083e-102)
   (/ (* x (/ (sin y) y)) z)
   (let* ((t_0 (/ y (sin y))))
     (if (<= x 1.6643543314697557e-7) (/ x (* z t_0)) (/ (/ 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 tmp;
	if (x <= -1.6559076628097083e-102) {
		tmp = (x * (sin(y) / y)) / z;
	} else {
		double t_0 = y / sin(y);
		double tmp_1;
		if (x <= 1.6643543314697557e-7) {
			tmp_1 = x / (z * t_0);
		} else {
			tmp_1 = (x / t_0) / 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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.65590766280970831e-102

    1. Initial program 0.9

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

    2. Applied *-commutative_binary640.9

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

    if -1.65590766280970831e-102 < x < 1.6643543314697557e-7

    1. Initial program 5.6

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

    2. Applied clear-num_binary645.6

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

    3. Applied un-div-inv_binary645.6

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

    4. Applied associate-/l/_binary640.1

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

    if 1.6643543314697557e-7 < x

    1. Initial program 0.2

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

    2. Applied clear-num_binary640.3

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

    3. Applied un-div-inv_binary640.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6559076628097083 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 1.6643543314697557 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))