Average Error: 24.7 → 6.4
Time: 5.0s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\
\;\;\;\;-1 \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r259116 = x;
        double r259117 = y;
        double r259118 = r259116 * r259117;
        double r259119 = z;
        double r259120 = r259118 * r259119;
        double r259121 = r259119 * r259119;
        double r259122 = t;
        double r259123 = a;
        double r259124 = r259122 * r259123;
        double r259125 = r259121 - r259124;
        double r259126 = sqrt(r259125);
        double r259127 = r259120 / r259126;
        return r259127;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r259128 = z;
        double r259129 = -1.1315032243066199e+154;
        bool r259130 = r259128 <= r259129;
        double r259131 = -1.0;
        double r259132 = x;
        double r259133 = y;
        double r259134 = r259132 * r259133;
        double r259135 = r259131 * r259134;
        double r259136 = 1.292444441257148e+68;
        bool r259137 = r259128 <= r259136;
        double r259138 = r259128 * r259128;
        double r259139 = t;
        double r259140 = a;
        double r259141 = r259139 * r259140;
        double r259142 = r259138 - r259141;
        double r259143 = sqrt(r259142);
        double r259144 = r259128 / r259143;
        double r259145 = r259134 * r259144;
        double r259146 = r259137 ? r259145 : r259134;
        double r259147 = r259130 ? r259135 : r259146;
        return r259147;
}

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

Original24.7
Target7.5
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1315032243066199e+154

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 0.9

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

    if -1.1315032243066199e+154 < z < 1.292444441257148e+68

    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-frac9.3

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

    6. Simplified9.3

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

    if 1.292444441257148e+68 < z

    1. Initial program 39.8

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

    2. Taylor expanded around inf 2.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1315032243066199 \cdot 10^{154}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.2924444412571481 \cdot 10^{68}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))))