Average Error: 6.3 → 1.3
Time: 19.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.029149003138143098746309211499089505365 \cdot 10^{64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;a \le 5.11858259250166290585004263509549340784 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -4.029149003138143098746309211499089505365 \cdot 10^{64}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;a \le 5.11858259250166290585004263509549340784 \cdot 10^{-56}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r280680 = x;
        double r280681 = y;
        double r280682 = z;
        double r280683 = t;
        double r280684 = r280682 - r280683;
        double r280685 = r280681 * r280684;
        double r280686 = a;
        double r280687 = r280685 / r280686;
        double r280688 = r280680 + r280687;
        return r280688;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r280689 = a;
        double r280690 = -4.029149003138143e+64;
        bool r280691 = r280689 <= r280690;
        double r280692 = y;
        double r280693 = r280692 / r280689;
        double r280694 = z;
        double r280695 = t;
        double r280696 = r280694 - r280695;
        double r280697 = x;
        double r280698 = fma(r280693, r280696, r280697);
        double r280699 = 5.118582592501663e-56;
        bool r280700 = r280689 <= r280699;
        double r280701 = 1.0;
        double r280702 = r280701 / r280689;
        double r280703 = r280696 * r280692;
        double r280704 = r280702 * r280703;
        double r280705 = r280704 + r280697;
        double r280706 = r280696 / r280689;
        double r280707 = fma(r280706, r280692, r280697);
        double r280708 = r280700 ? r280705 : r280707;
        double r280709 = r280691 ? r280698 : r280708;
        return r280709;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.6
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -4.029149003138143e+64

    1. Initial program 11.4

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

    2. Simplified2.2

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

    if -4.029149003138143e+64 < a < 5.118582592501663e-56

    1. Initial program 1.2

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

    2. Simplified3.6

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

    4. Applied fma-udef3.6

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

    5. Simplified3.2

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

    7. Applied div-inv3.3

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

    8. Applied *-un-lft-identity3.3

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

    9. Applied times-frac1.4

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

    10. Simplified1.3

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

    if 5.118582592501663e-56 < a

    1. Initial program 8.1

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

    2. Simplified1.8

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

    4. Applied fma-udef1.8

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

    5. Simplified2.0

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

    7. Applied *-un-lft-identity2.0

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

    8. Applied *-un-lft-identity2.0

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

    9. Applied distribute-lft-out2.0

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

    10. Simplified0.7

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.029149003138143098746309211499089505365 \cdot 10^{64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;a \le 5.11858259250166290585004263509549340784 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))