Average Error: 3.5 → 0.4
Time: 4.3s
Precision: binary64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
↓
\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -545059296815.9845:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 5.132963417392212 \cdot 10^{-49}:\\ \;\;\;\;t_1 + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array} \]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
t_1 := x - \frac{y}{z \cdot 3}\\
\mathbf{if}\;z \cdot 3 \leq -545059296815.9845:\\
\;\;\;\;t_1 + \frac{t}{3 \cdot \left(z \cdot y\right)}\\

\mathbf{elif}\;z \cdot 3 \leq 5.132963417392212 \cdot 10^{-49}:\\
\;\;\;\;t_1 + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\

\mathbf{else}:\\
\;\;\;\;t_1 + \frac{t}{z \cdot \left(3 \cdot y\right)}\\


\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
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (* z 3.0) -545059296815.9845)
     (+ t_1 (/ t (* 3.0 (* z y))))
     (if (<= (* z 3.0) 5.132963417392212e-49)
       (+ t_1 (* (/ 1.0 (* z 3.0)) (/ t y)))
       (+ t_1 (/ t (* z (* 3.0 y))))))))
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 t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((z * 3.0) <= -545059296815.9845) {
		tmp = t_1 + (t / (3.0 * (z * y)));
	} else if ((z * 3.0) <= 5.132963417392212e-49) {
		tmp = t_1 + ((1.0 / (z * 3.0)) * (t / y));
	} else {
		tmp = t_1 + (t / (z * (3.0 * y)));
	}
	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

Original3.5
Target1.8
Herbie0.4
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z 3) < -545059296815.9845

    1. Initial program 0.4

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

    2. Taylor expanded in z around 0 0.4

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

    if -545059296815.9845 < (*.f64 z 3) < 5.13296341739221169e-49

    1. Initial program 10.9

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

    2. Applied *-un-lft-identity_binary6410.9

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

    3. Applied times-frac_binary640.3

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

    if 5.13296341739221169e-49 < (*.f64 z 3)

    1. Initial program 0.5

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

    2. Applied associate-*l*_binary640.5

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

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -545059296815.9845:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 5.132963417392212 \cdot 10^{-49}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))