Average Error: 6.4 → 5.4
Time: 6.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{y}{\frac{a}{z - t}}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{y}{\frac{a}{z - t}}
double f(double x, double y, double z, double t, double a) {
        double r241083 = x;
        double r241084 = y;
        double r241085 = z;
        double r241086 = t;
        double r241087 = r241085 - r241086;
        double r241088 = r241084 * r241087;
        double r241089 = a;
        double r241090 = r241088 / r241089;
        double r241091 = r241083 + r241090;
        return r241091;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r241092 = x;
        double r241093 = y;
        double r241094 = a;
        double r241095 = z;
        double r241096 = t;
        double r241097 = r241095 - r241096;
        double r241098 = r241094 / r241097;
        double r241099 = r241093 / r241098;
        double r241100 = r241092 + r241099;
        return r241100;
}

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

Original6.4
Target0.8
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7842927315890493e+29

    1. Initial program 17.3

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

    3. Applied associate-/l*1.1

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

    5. Applied associate-/r/4.4

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

    7. Applied add-cube-cbrt5.0

      \[\leadsto x + \frac{y}{a} \cdot \color{blue}{\left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}\right)}\]

    8. Applied associate-*r*5.0

      \[\leadsto x + \color{blue}{\left(\frac{y}{a} \cdot \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)\right) \cdot \sqrt[3]{z - t}}\]

    9. Taylor expanded around 0 17.3

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

    10. Simplified0.9

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

    if -1.7842927315890493e+29 < y < 8.192485547752354e-59

    1. Initial program 0.6

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

    3. Applied sub-neg0.6

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

    4. Applied distribute-lft-in0.6

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

    if 8.192485547752354e-59 < y

    1. Initial program 12.4

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

    3. Applied associate-/l*1.1

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

  4. Final simplification5.4

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

Reproduce

herbie shell --seed 1978988140 
(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)))