Average Error: 2.6 → 2.1
Time: 3.6s
Precision: binary64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq 4.5577939887046744 \cdot 10^{-133}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \left(t \cdot \frac{1}{y}\right) \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \end{array} \]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;t \leq 4.5577939887046744 \cdot 10^{-133}:\\
\;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \left(t \cdot \frac{1}{y}\right) \cdot \frac{0.3333333333333333}{z}\\

\mathbf{else}:\\
\;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + t \cdot \frac{0.3333333333333333}{y \cdot z}\\


\end{array}

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.5577939887046744e-133)
   (+ (- x (/ y (* z 3.0))) (* (* t (/ 1.0 y)) (/ 0.3333333333333333 z)))
   (+ (- x (/ (/ y z) 3.0)) (* t (/ 0.3333333333333333 (* y z))))))
double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.5577939887046744e-133) {
		tmp = (x - (y / (z * 3.0))) + ((t * (1.0 / y)) * (0.3333333333333333 / z));
	} else {
		tmp = (x - ((y / z) / 3.0)) + (t * (0.3333333333333333 / (y * z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.6
Target2.6
Herbie2.1
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

  1. Split input into 2 regimes

  2. if t < 4.55779398870467442e-133

    1. Initial program 3.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

    2. Applied div-inv_binary643.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}} \]

    3. Simplified3.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + t \cdot \color{blue}{\frac{0.3333333333333333}{y \cdot z}} \]

    4. Applied *-un-lft-identity_binary643.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{\color{blue}{1 \cdot 0.3333333333333333}}{y \cdot z} \]

    5. Applied times-frac_binary643.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + t \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{0.3333333333333333}{z}\right)} \]

    6. Applied associate-*r*_binary642.3

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\left(t \cdot \frac{1}{y}\right) \cdot \frac{0.3333333333333333}{z}} \]

    if 4.55779398870467442e-133 < t

    1. Initial program 1.5

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

    2. Applied div-inv_binary641.7

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}} \]

    3. Simplified1.6

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + t \cdot \color{blue}{\frac{0.3333333333333333}{y \cdot z}} \]

    4. Applied associate-/r*_binary641.6

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + t \cdot \frac{0.3333333333333333}{y \cdot z} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5577939887046744 \cdot 10^{-133}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \left(t \cdot \frac{1}{y}\right) \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))