Average Error: 6.0 → 1.2
Time: 13.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \mathbf{elif}\;a \le 1.669828349187371898605130590820026774835 \cdot 10^{-121}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;a \le 1.540672155547814806102531356141137000031 \cdot 10^{96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\
\;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\

\mathbf{elif}\;a \le 1.669828349187371898605130590820026774835 \cdot 10^{-121}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{elif}\;a \le 1.540672155547814806102531356141137000031 \cdot 10^{96}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{y \cdot z}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r278152 = x;
        double r278153 = y;
        double r278154 = z;
        double r278155 = t;
        double r278156 = r278154 - r278155;
        double r278157 = r278153 * r278156;
        double r278158 = a;
        double r278159 = r278157 / r278158;
        double r278160 = r278152 - r278159;
        return r278160;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r278161 = a;
        double r278162 = -1.5982814066709494e-10;
        bool r278163 = r278161 <= r278162;
        double r278164 = y;
        double r278165 = t;
        double r278166 = z;
        double r278167 = r278165 - r278166;
        double r278168 = r278161 / r278167;
        double r278169 = r278164 / r278168;
        double r278170 = x;
        double r278171 = r278169 + r278170;
        double r278172 = 1.669828349187372e-121;
        bool r278173 = r278161 <= r278172;
        double r278174 = r278166 - r278165;
        double r278175 = r278164 * r278174;
        double r278176 = r278175 / r278161;
        double r278177 = r278170 - r278176;
        double r278178 = 1.5406721555478148e+96;
        bool r278179 = r278161 <= r278178;
        double r278180 = r278165 / r278161;
        double r278181 = fma(r278180, r278164, r278170);
        double r278182 = r278164 * r278166;
        double r278183 = r278182 / r278161;
        double r278184 = r278181 - r278183;
        double r278185 = r278179 ? r278184 : r278171;
        double r278186 = r278173 ? r278177 : r278185;
        double r278187 = r278163 ? r278171 : r278186;
        return r278187;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.5982814066709494e-10 or 1.5406721555478148e+96 < a

    1. Initial program 9.8

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

    2. Simplified1.7

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

    4. Applied fma-udef1.7

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

    5. Simplified0.5

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

    if -1.5982814066709494e-10 < a < 1.669828349187372e-121

    1. Initial program 1.1

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

    if 1.669828349187372e-121 < a < 1.5406721555478148e+96

    1. Initial program 2.2

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

    2. Simplified1.0

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

    4. Applied fma-udef1.0

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

    5. Simplified3.4

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

    6. Taylor expanded around 0 2.2

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

    7. Simplified3.5

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \mathbf{elif}\;a \le 1.669828349187371898605130590820026774835 \cdot 10^{-121}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;a \le 1.540672155547814806102531356141137000031 \cdot 10^{96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right) - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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