Average Error: 6.2 → 1.4
Time: 2.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;a \le 160974762298423667326976:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r259083 = x;
        double r259084 = y;
        double r259085 = z;
        double r259086 = t;
        double r259087 = r259085 - r259086;
        double r259088 = r259084 * r259087;
        double r259089 = a;
        double r259090 = r259088 / r259089;
        double r259091 = r259083 - r259090;
        return r259091;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r259092 = a;
        double r259093 = -9.123265245159162e-287;
        bool r259094 = r259092 <= r259093;
        double r259095 = y;
        double r259096 = r259095 / r259092;
        double r259097 = t;
        double r259098 = z;
        double r259099 = r259097 - r259098;
        double r259100 = x;
        double r259101 = fma(r259096, r259099, r259100);
        double r259102 = 1.6097476229842367e+23;
        bool r259103 = r259092 <= r259102;
        double r259104 = 1.0;
        double r259105 = r259098 - r259097;
        double r259106 = r259095 * r259105;
        double r259107 = r259092 / r259106;
        double r259108 = r259104 / r259107;
        double r259109 = r259100 - r259108;
        double r259110 = sqrt(r259092);
        double r259111 = r259095 / r259110;
        double r259112 = r259105 / r259110;
        double r259113 = r259111 * r259112;
        double r259114 = r259100 - r259113;
        double r259115 = r259103 ? r259109 : r259114;
        double r259116 = r259094 ? r259101 : r259115;
        return r259116;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -9.123265245159162e-287

    1. Initial program 6.4

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

    2. Simplified2.0

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

    if -9.123265245159162e-287 < a < 1.6097476229842367e+23

    1. Initial program 1.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied clear-num1.0

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

    if 1.6097476229842367e+23 < a

    1. Initial program 10.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt10.1

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

    4. Applied times-frac0.7

      \[\leadsto x - \color{blue}{\frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))