Average Error: 3.7 → 1.3
Time: 19.0s
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 r529186 = x;
        double r529187 = y;
        double r529188 = z;
        double r529189 = 3.0;
        double r529190 = r529188 * r529189;
        double r529191 = r529187 / r529190;
        double r529192 = r529186 - r529191;
        double r529193 = t;
        double r529194 = r529190 * r529187;
        double r529195 = r529193 / r529194;
        double r529196 = r529192 + r529195;
        return r529196;
}
↓
double f(double x, double y, double z, double t) {
        double r529197 = t;
        double r529198 = 1.0180250941212652e-108;
        bool r529199 = r529197 <= r529198;
        double r529200 = x;
        double r529201 = y;
        double r529202 = z;
        double r529203 = r529201 / r529202;
        double r529204 = 3.0;
        double r529205 = r529203 / r529204;
        double r529206 = r529200 - r529205;
        double r529207 = cbrt(r529197);
        double r529208 = r529207 * r529207;
        double r529209 = r529208 / r529202;
        double r529210 = r529207 / r529204;
        double r529211 = r529210 / r529201;
        double r529212 = r529209 * r529211;
        double r529213 = r529206 + r529212;
        double r529214 = r529202 * r529204;
        double r529215 = r529201 / r529214;
        double r529216 = r529200 - r529215;
        double r529217 = r529204 * r529201;
        double r529218 = r529202 * r529217;
        double r529219 = r529197 / r529218;
        double r529220 = r529216 + r529219;
        double r529221 = r529199 ? r529213 : r529220;
        return r529221;
}

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 +o rules:numerics
(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))))