Average Error: 2.8 → 1.4
Time: 6.5s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 2.6612179024617 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}} \cdot \frac{x}{\frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 2.6612179024617 \cdot 10^{-316}:\\
\;\;\;\;\frac{1}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}} \cdot \frac{x}{\frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x (/ (sin y) y)) 2.6612179024617e-316)
   (*
    (/ 1.0 (/ 1.0 (* (cbrt (/ (sin y) y)) (cbrt (/ (sin y) y)))))
    (/ x (/ z (cbrt (/ (sin y) y)))))
   (/ (* x (/ 1.0 (/ 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 tmp;
	if ((x * (sin(y) / y)) <= 2.6612179024617e-316) {
		tmp = (1.0 / (1.0 / (cbrt(sin(y) / y) * cbrt(sin(y) / y)))) * (x / (z / cbrt(sin(y) / y)));
	} else {
		tmp = (x * (1.0 / (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

Original2.8
Target0.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 2.661217902e-316

    1. Initial program 4.7

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary64_146862.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_147762.8

      \[\leadsto \frac{x}{\frac{z}{\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}\right) \cdot \sqrt[3]{\frac{\sin y}{y}}}}}\]

    6. Applied *-un-lft-identity_binary64_147412.8

      \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}\right) \cdot \sqrt[3]{\frac{\sin y}{y}}}}\]

    7. Applied times-frac_binary64_147472.8

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}} \cdot \frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}}\]

    8. Applied *-un-lft-identity_binary64_147412.8

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}} \cdot \frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}\]

    9. Applied times-frac_binary64_147472.4

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}} \cdot \frac{x}{\frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}}\]

    if 2.661217902e-316 < (*.f64 x (/.f64 (sin.f64 y) y))

    1. Initial program 0.2

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
    2. Using strategy rm

    3. Applied clear-num_binary64_147400.2

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 2.6612179024617 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}} \cdot \frac{x}{\frac{z}{\sqrt[3]{\frac{\sin y}{y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))