Average Error: 6.1 → 1.2
Time: 19.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.1628776471159573 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\ \mathbf{elif}\;a \le 1.7350666396159628 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.1628776471159573 \cdot 10^{-32}:\\
\;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\

\mathbf{elif}\;a \le 1.7350666396159628 \cdot 10^{+45}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r14623284 = x;
        double r14623285 = y;
        double r14623286 = z;
        double r14623287 = t;
        double r14623288 = r14623286 - r14623287;
        double r14623289 = r14623285 * r14623288;
        double r14623290 = a;
        double r14623291 = r14623289 / r14623290;
        double r14623292 = r14623284 + r14623291;
        return r14623292;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14623293 = a;
        double r14623294 = -2.1628776471159573e-32;
        bool r14623295 = r14623293 <= r14623294;
        double r14623296 = y;
        double r14623297 = r14623296 / r14623293;
        double r14623298 = z;
        double r14623299 = t;
        double r14623300 = r14623298 - r14623299;
        double r14623301 = r14623297 * r14623300;
        double r14623302 = x;
        double r14623303 = r14623301 + r14623302;
        double r14623304 = 1.7350666396159628e+45;
        bool r14623305 = r14623293 <= r14623304;
        double r14623306 = r14623300 * r14623296;
        double r14623307 = r14623306 / r14623293;
        double r14623308 = r14623307 + r14623302;
        double r14623309 = sqrt(r14623293);
        double r14623310 = r14623296 / r14623309;
        double r14623311 = r14623300 / r14623309;
        double r14623312 = r14623310 * r14623311;
        double r14623313 = r14623312 + r14623302;
        double r14623314 = r14623305 ? r14623308 : r14623313;
        double r14623315 = r14623295 ? r14623303 : r14623314;
        return r14623315;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target0.6
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -2.1628776471159573e-32

    1. Initial program 8.7

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

    2. Simplified1.6

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

    4. Applied fma-udef1.6

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

    if -2.1628776471159573e-32 < a < 1.7350666396159628e+45

    1. Initial program 0.8

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

    2. Simplified3.5

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

    4. Applied fma-udef3.5

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

    6. Applied associate-*r/0.8

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

    if 1.7350666396159628e+45 < a

    1. Initial program 10.6

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

    2. Simplified2.1

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

    4. Applied fma-udef2.1

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

    6. Applied add-sqr-sqrt2.3

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

    7. Applied *-un-lft-identity2.3

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

    8. Applied times-frac2.3

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

    9. Applied associate-*r*1.1

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

    10. Simplified1.0

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.1628776471159573 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\ \mathbf{elif}\;a \le 1.7350666396159628 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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