Average Error: 2.7 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -6.671106624591831 \cdot 10^{-178}:\\ \;\;\;\;\frac{t_1}{z}\\ \mathbf{elif}\;t_1 \leq 1.1284194523242232 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
t_0 := \frac{\sin y}{y}\\
t_1 := x \cdot t_0\\
\mathbf{if}\;t_1 \leq -6.671106624591831 \cdot 10^{-178}:\\
\;\;\;\;\frac{t_1}{z}\\

\mathbf{elif}\;t_1 \leq 1.1284194523242232 \cdot 10^{-223}:\\
\;\;\;\;t_0 \cdot \frac{x}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{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)) (t_1 (* x t_0)))
   (if (<= t_1 -6.671106624591831e-178)
     (/ t_1 z)
     (if (<= t_1 1.1284194523242232e-223)
       (* t_0 (/ x z))
       (/ (/ x (/ y (sin y))) 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 <= -6.671106624591831e-178) {
		tmp = t_1 / z;
	} else if (t_1 <= 1.1284194523242232e-223) {
		tmp = t_0 * (x / z);
	} else {
		tmp = (x / (y / sin(y))) / 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

Original2.7
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 3 regimes

  2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -6.6711066245918307e-178

    1. Initial program 0.2

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

    if -6.6711066245918307e-178 < (*.f64 x (/.f64 (sin.f64 y) y)) < 1.12841945232422318e-223

    1. Initial program 8.0

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

    2. Applied associate-/l*_binary640.5

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

    3. Applied associate-/r/_binary640.4

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

    if 1.12841945232422318e-223 < (*.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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -6.671106624591831 \cdot 10^{-178}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 1.1284194523242232 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))