Average Error: 24.8 → 7.3
Time: 4.8s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.7050069327788126 \cdot 10^{76}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 6.2711743111954205 \cdot 10^{136}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.7050069327788126 \cdot 10^{76}:\\
\;\;\;\;\left(x \cdot y\right) \cdot -1\\

\mathbf{elif}\;z \le 6.2711743111954205 \cdot 10^{136}:\\
\;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r233231 = x;
        double r233232 = y;
        double r233233 = r233231 * r233232;
        double r233234 = z;
        double r233235 = r233233 * r233234;
        double r233236 = r233234 * r233234;
        double r233237 = t;
        double r233238 = a;
        double r233239 = r233237 * r233238;
        double r233240 = r233236 - r233239;
        double r233241 = sqrt(r233240);
        double r233242 = r233235 / r233241;
        return r233242;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r233243 = z;
        double r233244 = -6.705006932778813e+76;
        bool r233245 = r233243 <= r233244;
        double r233246 = x;
        double r233247 = y;
        double r233248 = r233246 * r233247;
        double r233249 = -1.0;
        double r233250 = r233248 * r233249;
        double r233251 = 6.27117431119542e+136;
        bool r233252 = r233243 <= r233251;
        double r233253 = r233247 * r233243;
        double r233254 = 1.0;
        double r233255 = r233243 * r233243;
        double r233256 = t;
        double r233257 = a;
        double r233258 = r233256 * r233257;
        double r233259 = r233255 - r233258;
        double r233260 = sqrt(r233259);
        double r233261 = r233254 / r233260;
        double r233262 = r233253 * r233261;
        double r233263 = r233246 * r233262;
        double r233264 = r233252 ? r233263 : r233248;
        double r233265 = r233245 ? r233250 : r233264;
        return r233265;
}

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

Original24.8
Target7.9
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.705006932778813e+76

    1. Initial program 41.0

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

    3. Applied *-un-lft-identity41.0

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

    4. Applied sqrt-prod41.0

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

    5. Applied times-frac38.1

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

    6. Simplified38.1

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

    7. Taylor expanded around -inf 2.9

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

    if -6.705006932778813e+76 < z < 6.27117431119542e+136

    1. Initial program 11.0

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

    3. Applied *-un-lft-identity11.0

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

    4. Applied sqrt-prod11.0

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

    5. Applied times-frac9.4

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

    6. Simplified9.4

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

    8. Applied associate-*l*9.2

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

    10. Applied div-inv9.3

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

    11. Applied associate-*r*10.6

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

    if 6.27117431119542e+136 < z

    1. Initial program 50.5

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.7050069327788126 \cdot 10^{76}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 6.2711743111954205 \cdot 10^{136}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))))