Average Error: 6.0 → 0.6
Time: 4.4s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.2076128670123546 \cdot 10^{171}:\\ \;\;\;\;x - y \cdot \left(\frac{z}{a} - \frac{t}{a}\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -9.2076128670123546 \cdot 10^{171}:\\
\;\;\;\;x - y \cdot \left(\frac{z}{a} - \frac{t}{a}\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r371581 = x;
        double r371582 = y;
        double r371583 = z;
        double r371584 = t;
        double r371585 = r371583 - r371584;
        double r371586 = r371582 * r371585;
        double r371587 = a;
        double r371588 = r371586 / r371587;
        double r371589 = r371581 - r371588;
        return r371589;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r371590 = y;
        double r371591 = z;
        double r371592 = t;
        double r371593 = r371591 - r371592;
        double r371594 = r371590 * r371593;
        double r371595 = -9.207612867012355e+171;
        bool r371596 = r371594 <= r371595;
        double r371597 = x;
        double r371598 = a;
        double r371599 = r371591 / r371598;
        double r371600 = r371592 / r371598;
        double r371601 = r371599 - r371600;
        double r371602 = r371590 * r371601;
        double r371603 = r371597 - r371602;
        double r371604 = 2.8461840642724514e+164;
        bool r371605 = r371594 <= r371604;
        double r371606 = r371594 / r371598;
        double r371607 = r371597 - r371606;
        double r371608 = r371598 / r371593;
        double r371609 = r371590 / r371608;
        double r371610 = r371597 - r371609;
        double r371611 = r371605 ? r371607 : r371610;
        double r371612 = r371596 ? r371603 : r371611;
        return r371612;
}

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

Original6.0
Target0.7
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -9.207612867012355e+171

    1. Initial program 22.7

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

    3. Applied associate-/l*1.2

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

    5. Applied div-inv1.3

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

    6. Applied associate-/r*1.5

      \[\leadsto x - \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt2.2

      \[\leadsto x - \frac{\frac{y}{a}}{\frac{1}{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}}\]

    9. Applied add-sqr-sqrt2.2

      \[\leadsto x - \frac{\frac{y}{a}}{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}\]

    10. Applied times-frac2.2

      \[\leadsto x - \frac{\frac{y}{a}}{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\sqrt{1}}{\sqrt[3]{z - t}}}}\]

    11. Applied *-un-lft-identity2.2

      \[\leadsto x - \frac{\frac{y}{\color{blue}{1 \cdot a}}}{\frac{\sqrt{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\sqrt{1}}{\sqrt[3]{z - t}}}\]

    12. Applied *-un-lft-identity2.2

      \[\leadsto x - \frac{\frac{\color{blue}{1 \cdot y}}{1 \cdot a}}{\frac{\sqrt{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\sqrt{1}}{\sqrt[3]{z - t}}}\]

    13. Applied times-frac2.2

      \[\leadsto x - \frac{\color{blue}{\frac{1}{1} \cdot \frac{y}{a}}}{\frac{\sqrt{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\sqrt{1}}{\sqrt[3]{z - t}}}\]

    14. Applied times-frac2.2

      \[\leadsto x - \color{blue}{\frac{\frac{1}{1}}{\frac{\sqrt{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}} \cdot \frac{\frac{y}{a}}{\frac{\sqrt{1}}{\sqrt[3]{z - t}}}}\]

    15. Simplified2.2

      \[\leadsto x - \color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)} \cdot \frac{\frac{y}{a}}{\frac{\sqrt{1}}{\sqrt[3]{z - t}}}\]

    16. Simplified2.2

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

    17. Taylor expanded around 0 22.7

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

    18. Simplified1.4

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

    if -9.207612867012355e+171 < (* y (- z t)) < 2.8461840642724514e+164

    1. Initial program 0.4

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

    if 2.8461840642724514e+164 < (* y (- z t))

    1. Initial program 23.2

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

    3. Applied associate-/l*1.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.2076128670123546 \cdot 10^{171}:\\ \;\;\;\;x - y \cdot \left(\frac{z}{a} - \frac{t}{a}\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :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)))