Average Error: 25.5 → 5.7
Time: 6.7s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.30855613287552178 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 6.3074422229433126 \cdot 10^{112}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.30855613287552178 \cdot 10^{104}:\\
\;\;\;\;-x \cdot y\\

\mathbf{elif}\;z \le 6.3074422229433126 \cdot 10^{112}:\\
\;\;\;\;\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{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 r351107 = x;
        double r351108 = y;
        double r351109 = r351107 * r351108;
        double r351110 = z;
        double r351111 = r351109 * r351110;
        double r351112 = r351110 * r351110;
        double r351113 = t;
        double r351114 = a;
        double r351115 = r351113 * r351114;
        double r351116 = r351112 - r351115;
        double r351117 = sqrt(r351116);
        double r351118 = r351111 / r351117;
        return r351118;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r351119 = z;
        double r351120 = -3.308556132875522e+104;
        bool r351121 = r351119 <= r351120;
        double r351122 = x;
        double r351123 = y;
        double r351124 = r351122 * r351123;
        double r351125 = -r351124;
        double r351126 = 6.307442222943313e+112;
        bool r351127 = r351119 <= r351126;
        double r351128 = cbrt(r351119);
        double r351129 = r351128 * r351128;
        double r351130 = r351119 * r351119;
        double r351131 = t;
        double r351132 = a;
        double r351133 = r351131 * r351132;
        double r351134 = r351130 - r351133;
        double r351135 = cbrt(r351134);
        double r351136 = fabs(r351135);
        double r351137 = r351129 / r351136;
        double r351138 = r351123 * r351137;
        double r351139 = r351122 * r351138;
        double r351140 = sqrt(r351135);
        double r351141 = r351128 / r351140;
        double r351142 = r351139 * r351141;
        double r351143 = r351127 ? r351142 : r351124;
        double r351144 = r351121 ? r351125 : r351143;
        return r351144;
}

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

Original25.5
Target7.6
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.308556132875522e+104

    1. Initial program 44.2

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

    2. Taylor expanded around -inf 2.6

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

    3. Simplified2.6

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

    if -3.308556132875522e+104 < z < 6.307442222943313e+112

    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 *-un-lft-identity11.2

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

    4. Applied sqrt-prod11.2

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

    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

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

    8. Applied add-cube-cbrt9.8

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right) \cdot \sqrt[3]{z \cdot z - t \cdot a}}}}\]

    9. Applied sqrt-prod9.8

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\color{blue}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

    10. Applied add-cube-cbrt10.0

      \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

    11. Applied times-frac10.1

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\right)}\]

    12. Applied associate-*r*9.3

      \[\leadsto \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

    13. Simplified8.2

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

    if 6.307442222943313e+112 < z

    1. Initial program 46.4

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

    2. Taylor expanded around inf 2.0

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.30855613287552178 \cdot 10^{104}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \le 6.3074422229433126 \cdot 10^{112}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))