Average Error: 6.5 → 1.0
Time: 23.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.737630196529751538621117105919303400464 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{1}{\frac{\frac{a}{y}}{t}} - x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\
\;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.737630196529751538621117105919303400464 \cdot 10^{-50}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{1}{\frac{\frac{a}{y}}{t}} - x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r67180 = x;
        double r67181 = y;
        double r67182 = z;
        double r67183 = t;
        double r67184 = r67182 - r67183;
        double r67185 = r67181 * r67184;
        double r67186 = a;
        double r67187 = r67185 / r67186;
        double r67188 = r67180 + r67187;
        return r67188;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r67189 = y;
        double r67190 = z;
        double r67191 = t;
        double r67192 = r67190 - r67191;
        double r67193 = r67189 * r67192;
        double r67194 = a;
        double r67195 = r67193 / r67194;
        double r67196 = -inf.0;
        bool r67197 = r67195 <= r67196;
        double r67198 = r67194 / r67189;
        double r67199 = r67190 / r67198;
        double r67200 = r67191 / r67198;
        double r67201 = x;
        double r67202 = r67200 - r67201;
        double r67203 = r67199 - r67202;
        double r67204 = 4.737630196529752e-50;
        bool r67205 = r67195 <= r67204;
        double r67206 = r67201 + r67195;
        double r67207 = 1.0;
        double r67208 = r67198 / r67191;
        double r67209 = r67207 / r67208;
        double r67210 = r67209 - r67201;
        double r67211 = r67199 - r67210;
        double r67212 = r67205 ? r67206 : r67211;
        double r67213 = r67197 ? r67203 : r67212;
        return r67213;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target0.7
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) a) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef0.2

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

    5. Simplified0.2

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]
    6. Using strategy rm

    7. Applied div-sub0.2

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

    8. Applied associate-+l-0.2

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

    if -inf.0 < (/ (* y (- z t)) a) < 4.737630196529752e-50

    1. Initial program 0.5

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

    if 4.737630196529752e-50 < (/ (* y (- z t)) a)

    1. Initial program 0.2

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

    2. Simplified3.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef3.0

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

    5. Simplified2.8

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]
    6. Using strategy rm

    7. Applied div-sub2.8

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

    8. Applied associate-+l-2.8

      \[\leadsto \color{blue}{\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)}\]
    9. Using strategy rm

    10. Applied clear-num2.8

      \[\leadsto \frac{z}{\frac{a}{y}} - \left(\color{blue}{\frac{1}{\frac{\frac{a}{y}}{t}}} - x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.737630196529751538621117105919303400464 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{1}{\frac{\frac{a}{y}}{t}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019310 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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