Average Error: 6.2 → 1.3
Time: 4.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -5.8671698741040733 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 1.2727793494611607 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \frac{\left(t - z\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -5.8671698741040733 \cdot 10^{-19}:\\
\;\;\;\;y \cdot \frac{t - z}{a} + x\\

\mathbf{elif}\;a \le 1.2727793494611607 \cdot 10^{53}:\\
\;\;\;\;1 \cdot \frac{\left(t - z\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r272141 = x;
        double r272142 = y;
        double r272143 = z;
        double r272144 = t;
        double r272145 = r272143 - r272144;
        double r272146 = r272142 * r272145;
        double r272147 = a;
        double r272148 = r272146 / r272147;
        double r272149 = r272141 - r272148;
        return r272149;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r272150 = a;
        double r272151 = -5.867169874104073e-19;
        bool r272152 = r272150 <= r272151;
        double r272153 = y;
        double r272154 = t;
        double r272155 = z;
        double r272156 = r272154 - r272155;
        double r272157 = r272156 / r272150;
        double r272158 = r272153 * r272157;
        double r272159 = x;
        double r272160 = r272158 + r272159;
        double r272161 = 1.2727793494611607e+53;
        bool r272162 = r272150 <= r272161;
        double r272163 = 1.0;
        double r272164 = r272156 * r272153;
        double r272165 = r272164 / r272150;
        double r272166 = r272163 * r272165;
        double r272167 = r272166 + r272159;
        double r272168 = r272153 / r272150;
        double r272169 = fma(r272168, r272156, r272159);
        double r272170 = r272162 ? r272167 : r272169;
        double r272171 = r272152 ? r272160 : r272170;
        return r272171;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.8
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -5.867169874104073e-19

    1. Initial program 9.0

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

    2. Simplified1.6

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

    4. Applied fma-udef1.6

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

    6. Applied div-inv1.6

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

    7. Applied associate-*l*0.6

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

    8. Simplified0.6

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

    if -5.867169874104073e-19 < a < 1.2727793494611607e+53

    1. Initial program 1.3

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

    2. Simplified3.4

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

    4. Applied fma-udef3.4

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

    6. Applied *-un-lft-identity3.4

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

    7. Applied *-un-lft-identity3.4

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

    8. Applied times-frac3.4

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

    9. Applied associate-*l*3.4

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

    10. Simplified1.3

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

    if 1.2727793494611607e+53 < a

    1. Initial program 10.3

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

    2. Simplified2.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.8671698741040733 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 1.2727793494611607 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \frac{\left(t - z\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))