Average Error: 6.8 → 0.8
Time: 2.4s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\
\;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\
\;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r339445 = x;
        double r339446 = y;
        double r339447 = z;
        double r339448 = r339447 - r339445;
        double r339449 = r339446 * r339448;
        double r339450 = t;
        double r339451 = r339449 / r339450;
        double r339452 = r339445 + r339451;
        return r339452;
}
↓
double f(double x, double y, double z, double t) {
        double r339453 = x;
        double r339454 = y;
        double r339455 = z;
        double r339456 = r339455 - r339453;
        double r339457 = r339454 * r339456;
        double r339458 = t;
        double r339459 = r339457 / r339458;
        double r339460 = r339453 + r339459;
        double r339461 = -inf.0;
        bool r339462 = r339460 <= r339461;
        double r339463 = r339458 / r339456;
        double r339464 = r339454 / r339463;
        double r339465 = r339453 + r339464;
        double r339466 = -9.694887903047778e+55;
        bool r339467 = r339460 <= r339466;
        double r339468 = 1.0;
        double r339469 = r339458 / r339457;
        double r339470 = r339468 / r339469;
        double r339471 = r339453 + r339470;
        double r339472 = 8.050245014478289e-243;
        bool r339473 = r339460 <= r339472;
        double r339474 = r339454 / r339458;
        double r339475 = fma(r339474, r339456, r339453);
        double r339476 = 5.670086864606111e+271;
        bool r339477 = r339460 <= r339476;
        double r339478 = r339477 ? r339460 : r339475;
        double r339479 = r339473 ? r339475 : r339478;
        double r339480 = r339467 ? r339471 : r339479;
        double r339481 = r339462 ? r339465 : r339480;
        return r339481;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.8
Target2.1
Herbie0.8
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 4 regimes

  2. if (+ x (/ (* y (- z x)) t)) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-/l*0.2

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

    if -inf.0 < (+ x (/ (* y (- z x)) t)) < -9.694887903047778e+55

    1. Initial program 0.1

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

    3. Applied clear-num0.2

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

    if -9.694887903047778e+55 < (+ x (/ (* y (- z x)) t)) < 8.050245014478289e-243 or 5.670086864606111e+271 < (+ x (/ (* y (- z x)) t))

    1. Initial program 11.8

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

    2. Simplified1.7

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

    if 8.050245014478289e-243 < (+ x (/ (* y (- z x)) t)) < 5.670086864606111e+271

    1. Initial program 0.6

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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