Average Error: 24.5 → 6.6
Time: 6.6s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.21372963348103654 \cdot 10^{154}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 8.84000957203954817 \cdot 10^{95}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.21372963348103654 \cdot 10^{154}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \le 8.84000957203954817 \cdot 10^{95}:\\
\;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\

\mathbf{else}:\\
\;\;\;\;y \cdot x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r264413 = x;
        double r264414 = y;
        double r264415 = r264413 * r264414;
        double r264416 = z;
        double r264417 = r264415 * r264416;
        double r264418 = r264416 * r264416;
        double r264419 = t;
        double r264420 = a;
        double r264421 = r264419 * r264420;
        double r264422 = r264418 - r264421;
        double r264423 = sqrt(r264422);
        double r264424 = r264417 / r264423;
        return r264424;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r264425 = z;
        double r264426 = -1.2137296334810365e+154;
        bool r264427 = r264425 <= r264426;
        double r264428 = y;
        double r264429 = x;
        double r264430 = r264428 * r264429;
        double r264431 = -r264430;
        double r264432 = 8.840009572039548e+95;
        bool r264433 = r264425 <= r264432;
        double r264434 = r264425 * r264425;
        double r264435 = t;
        double r264436 = a;
        double r264437 = r264435 * r264436;
        double r264438 = r264434 - r264437;
        double r264439 = sqrt(r264438);
        double r264440 = r264425 / r264439;
        double r264441 = r264429 * r264440;
        double r264442 = r264428 * r264441;
        double r264443 = r264433 ? r264442 : r264430;
        double r264444 = r264427 ? r264431 : r264443;
        return r264444;
}

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

Original24.5
Target8.0
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.2137296334810365e+154

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

      \[\leadsto \color{blue}{-y \cdot x}\]

    if -1.2137296334810365e+154 < z < 8.840009572039548e+95

    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-frac8.8

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

    6. Simplified8.8

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

    8. Applied associate-*l*9.0

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

    if 8.840009572039548e+95 < z

    1. Initial program 43.1

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

    2. Taylor expanded around inf 3.0

      \[\leadsto \color{blue}{x \cdot y}\]

    3. Simplified3.0

      \[\leadsto \color{blue}{y \cdot x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.21372963348103654 \cdot 10^{154}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 8.84000957203954817 \cdot 10^{95}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))