Average Error: 24.2 → 6.2
Time: 8.6s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.6183547901220424 \cdot 10^{153}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 6.9260668882011419 \cdot 10^{147}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.6183547901220424 \cdot 10^{153}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \le 6.9260668882011419 \cdot 10^{147}:\\
\;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r294653 = x;
        double r294654 = y;
        double r294655 = r294653 * r294654;
        double r294656 = z;
        double r294657 = r294655 * r294656;
        double r294658 = r294656 * r294656;
        double r294659 = t;
        double r294660 = a;
        double r294661 = r294659 * r294660;
        double r294662 = r294658 - r294661;
        double r294663 = sqrt(r294662);
        double r294664 = r294657 / r294663;
        return r294664;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r294665 = z;
        double r294666 = -8.618354790122042e+153;
        bool r294667 = r294665 <= r294666;
        double r294668 = y;
        double r294669 = x;
        double r294670 = r294668 * r294669;
        double r294671 = -r294670;
        double r294672 = 6.926066888201142e+147;
        bool r294673 = r294665 <= r294672;
        double r294674 = r294665 * r294665;
        double r294675 = t;
        double r294676 = a;
        double r294677 = r294675 * r294676;
        double r294678 = r294674 - r294677;
        double r294679 = sqrt(r294678);
        double r294680 = r294679 / r294665;
        double r294681 = r294668 / r294680;
        double r294682 = r294669 * r294681;
        double r294683 = r294673 ? r294682 : r294670;
        double r294684 = r294667 ? r294671 : r294683;
        return r294684;
}

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
Target7.7
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -8.618354790122042e+153

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 1.1

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

    3. Simplified1.1

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

    if -8.618354790122042e+153 < z < 6.926066888201142e+147

    1. Initial program 11.2

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

    3. Applied associate-/l*8.9

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

    5. Applied *-un-lft-identity8.9

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

    6. Applied *-un-lft-identity8.9

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

    7. Applied sqrt-prod8.9

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

    8. Applied times-frac8.9

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

    9. Applied times-frac8.6

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

    10. Simplified8.6

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

    if 6.926066888201142e+147 < z

    1. Initial program 51.6

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

    2. Taylor expanded around inf 1.2

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

    3. Simplified1.2

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.6183547901220424 \cdot 10^{153}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 6.9260668882011419 \cdot 10^{147}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))))