Average Error: 24.2 → 6.9
Time: 4.3s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.20302089242684697669438190506894627496 \cdot 10^{85}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 5.834852428696666747363497161733577764251 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.20302089242684697669438190506894627496 \cdot 10^{85}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 5.834852428696666747363497161733577764251 \cdot 10^{125}:\\
\;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot y\right) \cdot 1\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r270304 = x;
        double r270305 = y;
        double r270306 = r270304 * r270305;
        double r270307 = z;
        double r270308 = r270306 * r270307;
        double r270309 = r270307 * r270307;
        double r270310 = t;
        double r270311 = a;
        double r270312 = r270310 * r270311;
        double r270313 = r270309 - r270312;
        double r270314 = sqrt(r270313);
        double r270315 = r270308 / r270314;
        return r270315;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r270316 = z;
        double r270317 = -1.203020892426847e+85;
        bool r270318 = r270316 <= r270317;
        double r270319 = -1.0;
        double r270320 = x;
        double r270321 = y;
        double r270322 = r270320 * r270321;
        double r270323 = r270319 * r270322;
        double r270324 = 5.834852428696667e+125;
        bool r270325 = r270316 <= r270324;
        double r270326 = r270321 * r270316;
        double r270327 = 1.0;
        double r270328 = r270316 * r270316;
        double r270329 = t;
        double r270330 = a;
        double r270331 = r270329 * r270330;
        double r270332 = r270328 - r270331;
        double r270333 = sqrt(r270332);
        double r270334 = r270327 / r270333;
        double r270335 = r270326 * r270334;
        double r270336 = r270320 * r270335;
        double r270337 = r270322 * r270327;
        double r270338 = r270325 ? r270336 : r270337;
        double r270339 = r270318 ? r270323 : r270338;
        return r270339;
}

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

Original24.2
Target7.5
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.203020892426847e+85

    1. Initial program 40.8

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

    2. Taylor expanded around -inf 2.9

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

    if -1.203020892426847e+85 < z < 5.834852428696667e+125

    1. Initial program 10.8

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

    3. Applied *-un-lft-identity10.8

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

    4. Applied sqrt-prod10.8

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

    5. Applied times-frac8.9

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

    6. Simplified8.9

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

    8. Applied associate-*l*8.4

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

    10. Applied div-inv8.5

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

    11. Applied associate-*r*10.0

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

    if 5.834852428696667e+125 < z

    1. Initial program 48.1

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

    3. Applied *-un-lft-identity48.1

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

    4. Applied sqrt-prod48.1

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

    5. Applied times-frac46.3

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

    6. Simplified46.3

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

    7. Taylor expanded around inf 1.5

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.20302089242684697669438190506894627496 \cdot 10^{85}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 5.834852428696666747363497161733577764251 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

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