Average Error: 6.4 → 0.7
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x - \frac{t}{\frac{a}{y}}\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\
\;\;\;\;y \cdot \frac{z - t}{a} + x\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r380195 = x;
        double r380196 = y;
        double r380197 = z;
        double r380198 = t;
        double r380199 = r380197 - r380198;
        double r380200 = r380196 * r380199;
        double r380201 = a;
        double r380202 = r380200 / r380201;
        double r380203 = r380195 + r380202;
        return r380203;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r380204 = y;
        double r380205 = z;
        double r380206 = t;
        double r380207 = r380205 - r380206;
        double r380208 = r380204 * r380207;
        double r380209 = -5.395750418806486e+136;
        bool r380210 = r380208 <= r380209;
        double r380211 = a;
        double r380212 = r380207 / r380211;
        double r380213 = r380204 * r380212;
        double r380214 = x;
        double r380215 = r380213 + r380214;
        double r380216 = 1.0672653342869649e+207;
        bool r380217 = r380208 <= r380216;
        double r380218 = r380208 / r380211;
        double r380219 = r380218 + r380214;
        double r380220 = r380204 / r380211;
        double r380221 = r380211 / r380204;
        double r380222 = r380206 / r380221;
        double r380223 = r380214 - r380222;
        double r380224 = fma(r380205, r380220, r380223);
        double r380225 = r380217 ? r380219 : r380224;
        double r380226 = r380210 ? r380215 : r380225;
        return r380226;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.4
Target0.7
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -5.395750418806486e+136

    1. Initial program 20.6

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

    2. Simplified1.5

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

    4. Applied fma-udef1.5

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

    6. Applied div-inv1.6

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

    7. Applied associate-*l*2.5

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

    8. Simplified2.4

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

    if -5.395750418806486e+136 < (* y (- z t)) < 1.0672653342869649e+207

    1. Initial program 0.4

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

    2. Simplified2.9

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

    4. Applied fma-udef2.9

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

    6. Applied associate-*l/0.4

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

    if 1.0672653342869649e+207 < (* y (- z t))

    1. Initial program 31.0

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

    2. Simplified0.4

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

    4. Applied fma-udef0.4

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

    6. Applied *-un-lft-identity0.4

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

    7. Applied *-un-lft-identity0.4

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

    8. Applied prod-diff0.4

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

    9. Applied distribute-lft-in0.4

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

    10. Applied associate-+l+0.4

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

    11. Simplified0.4

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

    12. Taylor expanded around 0 31.0

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

    13. Simplified17.4

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

    15. Applied associate-/l*0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.39575041880648583 \cdot 10^{136}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.0672653342869649 \cdot 10^{207}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x - \frac{t}{\frac{a}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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 -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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