Average Error: 6.0 → 0.8
Time: 18.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{elif}\;a \le 1.051524309559396278412548750425595537339 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\
\;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

\mathbf{elif}\;a \le 1.051524309559396278412548750425595537339 \cdot 10^{-47}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13168675 = x;
        double r13168676 = y;
        double r13168677 = z;
        double r13168678 = t;
        double r13168679 = r13168677 - r13168678;
        double r13168680 = r13168676 * r13168679;
        double r13168681 = a;
        double r13168682 = r13168680 / r13168681;
        double r13168683 = r13168675 + r13168682;
        return r13168683;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13168684 = a;
        double r13168685 = -1.5982814066709494e-10;
        bool r13168686 = r13168684 <= r13168685;
        double r13168687 = x;
        double r13168688 = y;
        double r13168689 = z;
        double r13168690 = t;
        double r13168691 = r13168689 - r13168690;
        double r13168692 = r13168684 / r13168691;
        double r13168693 = r13168688 / r13168692;
        double r13168694 = r13168687 + r13168693;
        double r13168695 = 6.452314596038024e-142;
        bool r13168696 = r13168684 <= r13168695;
        double r13168697 = r13168691 * r13168688;
        double r13168698 = r13168697 / r13168684;
        double r13168699 = r13168698 + r13168687;
        double r13168700 = 1.0515243095593963e-47;
        bool r13168701 = r13168684 <= r13168700;
        double r13168702 = r13168688 / r13168684;
        double r13168703 = fma(r13168702, r13168691, r13168687);
        double r13168704 = r13168691 / r13168684;
        double r13168705 = fma(r13168688, r13168704, r13168687);
        double r13168706 = r13168701 ? r13168703 : r13168705;
        double r13168707 = r13168696 ? r13168699 : r13168706;
        double r13168708 = r13168686 ? r13168694 : r13168707;
        return r13168708;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if a < -1.5982814066709494e-10

    1. Initial program 8.4

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

    3. Applied associate-/l*0.4

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

    if -1.5982814066709494e-10 < a < 6.452314596038024e-142

    1. Initial program 1.1

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

    if 6.452314596038024e-142 < a < 1.0515243095593963e-47

    1. Initial program 0.6

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

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

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

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

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

    5. Applied distribute-lft-out0.6

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

    6. Simplified2.2

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

    if 1.0515243095593963e-47 < a

    1. Initial program 8.7

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

    2. Simplified0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{elif}\;a \le 1.051524309559396278412548750425595537339 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]

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, E"

  :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)))