Average Error: 2.8 → 0.3
Time: 5.8s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.6572955623815812 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{y}{\sin y}\\ \mathbf{if}\;z \leq 1.0432202070371389 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{t_0}\\ \end{array}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.6572955623815812 \cdot 10^{-51}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{y}{\sin y}\\
\mathbf{if}\;z \leq 1.0432202070371389 \cdot 10^{-17}:\\
\;\;\;\;\frac{x}{z \cdot t_0}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{z} \cdot \frac{x}{t_0}\\


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6572955623815812e-51)
   (/ (/ x z) (/ 1.0 (/ (sin y) y)))
   (let* ((t_0 (/ y (sin y))))
     (if (<= z 1.0432202070371389e-17)
       (/ x (* z t_0))
       (* (/ 1.0 z) (/ x t_0))))))
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 (z <= -2.6572955623815812e-51) {
		tmp = (x / z) / (1.0 / (sin(y) / y));
	} else {
		double t_0 = y / sin(y);
		double tmp_1;
		if (z <= 1.0432202070371389e-17) {
			tmp_1 = x / (z * t_0);
		} else {
			tmp_1 = (1.0 / z) * (x / t_0);
		}
		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 z < -2.65729556238158123e-51

    1. Initial program 0.2

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

    2. Applied associate-/l*_binary644.9

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

    3. Applied div-inv_binary644.9

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

    4. Applied associate-/r*_binary640.3

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

    if -2.65729556238158123e-51 < z < 1.0432202070371389e-17

    1. Initial program 6.7

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

    2. Applied associate-/l*_binary640.2

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

    3. Applied div-inv_binary640.3

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

    4. Simplified0.3

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

    if 1.0432202070371389e-17 < z

    1. Initial program 0.1

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

    2. Applied associate-/l*_binary645.2

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

    3. Applied div-inv_binary645.2

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

    4. Simplified5.2

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

    5. Applied *-un-lft-identity_binary645.2

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

    6. Applied times-frac_binary640.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6572955623815812 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{elif}\;z \leq 1.0432202070371389 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y}{\sin y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))