Average Error: 3.7 → 1.3
Time: 12.8s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\
\;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r484178 = x;
        double r484179 = y;
        double r484180 = z;
        double r484181 = 3.0;
        double r484182 = r484180 * r484181;
        double r484183 = r484179 / r484182;
        double r484184 = r484178 - r484183;
        double r484185 = t;
        double r484186 = r484182 * r484179;
        double r484187 = r484185 / r484186;
        double r484188 = r484184 + r484187;
        return r484188;
}
↓
double f(double x, double y, double z, double t) {
        double r484189 = t;
        double r484190 = 1.0180250941212652e-108;
        bool r484191 = r484189 <= r484190;
        double r484192 = x;
        double r484193 = y;
        double r484194 = z;
        double r484195 = r484193 / r484194;
        double r484196 = 3.0;
        double r484197 = r484195 / r484196;
        double r484198 = r484192 - r484197;
        double r484199 = cbrt(r484189);
        double r484200 = r484199 * r484199;
        double r484201 = r484200 / r484194;
        double r484202 = r484199 / r484196;
        double r484203 = r484202 / r484193;
        double r484204 = r484201 * r484203;
        double r484205 = r484198 + r484204;
        double r484206 = r484194 * r484196;
        double r484207 = r484193 / r484206;
        double r484208 = r484192 - r484207;
        double r484209 = r484196 * r484193;
        double r484210 = r484194 * r484209;
        double r484211 = r484189 / r484210;
        double r484212 = r484208 + r484211;
        double r484213 = r484191 ? r484205 : r484212;
        return r484213;
}

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

Original3.7
Target1.8
Herbie1.3
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 1.0180250941212652e-108

    1. Initial program 4.9

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*1.8

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

    5. Applied associate-/r*1.8

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

    7. Applied *-un-lft-identity1.8

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

    8. Applied add-cube-cbrt2.1

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

    9. Applied times-frac2.0

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

    10. Applied times-frac1.4

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

    11. Simplified1.4

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

    if 1.0180250941212652e-108 < t

    1. Initial program 1.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-*l*1.2

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))