Average Error: 24.6 → 7.5
Time: 4.1s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\
\;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r287417 = x;
        double r287418 = y;
        double r287419 = r287417 * r287418;
        double r287420 = z;
        double r287421 = r287419 * r287420;
        double r287422 = r287420 * r287420;
        double r287423 = t;
        double r287424 = a;
        double r287425 = r287423 * r287424;
        double r287426 = r287422 - r287425;
        double r287427 = sqrt(r287426);
        double r287428 = r287421 / r287427;
        return r287428;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r287429 = z;
        double r287430 = -1.1453877187877288e+27;
        bool r287431 = r287429 <= r287430;
        double r287432 = -1.0;
        double r287433 = x;
        double r287434 = y;
        double r287435 = r287433 * r287434;
        double r287436 = r287432 * r287435;
        double r287437 = 1.6894597162167635e+31;
        bool r287438 = r287429 <= r287437;
        double r287439 = r287434 * r287429;
        double r287440 = r287433 * r287439;
        double r287441 = r287429 * r287429;
        double r287442 = t;
        double r287443 = a;
        double r287444 = r287442 * r287443;
        double r287445 = r287441 - r287444;
        double r287446 = sqrt(r287445);
        double r287447 = r287440 / r287446;
        double r287448 = 1.0;
        double r287449 = r287435 * r287448;
        double r287450 = r287438 ? r287447 : r287449;
        double r287451 = r287431 ? r287436 : r287450;
        return r287451;
}

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

Original24.6
Target7.6
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1453877187877288e+27

    1. Initial program 34.6

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

    3. Applied *-un-lft-identity34.6

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

    4. Applied sqrt-prod34.6

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

    5. Applied times-frac32.4

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

    6. Simplified32.4

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

    8. Applied fma-neg32.4

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

    9. Taylor expanded around -inf 3.9

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

    if -1.1453877187877288e+27 < z < 1.6894597162167635e+31

    1. Initial program 11.8

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

    3. Applied associate-*l*11.6

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

    if 1.6894597162167635e+31 < z

    1. Initial program 35.6

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

    3. Applied *-un-lft-identity35.6

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

    4. Applied sqrt-prod35.6

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

    5. Applied times-frac32.8

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

    6. Simplified32.8

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

    8. Applied fma-neg32.8

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

    9. Taylor expanded around inf 4.3

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.145387718787729 \cdot 10^{27}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.68945971621676345 \cdot 10^{31}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))))