Average Error: 24.3 → 5.3
Time: 18.5s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.718674623067115332199814933152861240084 \cdot 10^{135}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 2.130897152856615796932912078487397342387 \cdot 10^{121}:\\ \;\;\;\;\frac{x}{\frac{\left|\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right|}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.718674623067115332199814933152861240084 \cdot 10^{135}:\\
\;\;\;\;-y \cdot x\\

\mathbf{elif}\;z \le 2.130897152856615796932912078487397342387 \cdot 10^{121}:\\
\;\;\;\;\frac{x}{\frac{\left|\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right|}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r233072 = x;
        double r233073 = y;
        double r233074 = r233072 * r233073;
        double r233075 = z;
        double r233076 = r233074 * r233075;
        double r233077 = r233075 * r233075;
        double r233078 = t;
        double r233079 = a;
        double r233080 = r233078 * r233079;
        double r233081 = r233077 - r233080;
        double r233082 = sqrt(r233081);
        double r233083 = r233076 / r233082;
        return r233083;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r233084 = z;
        double r233085 = -2.7186746230671153e+135;
        bool r233086 = r233084 <= r233085;
        double r233087 = y;
        double r233088 = x;
        double r233089 = r233087 * r233088;
        double r233090 = -r233089;
        double r233091 = 2.130897152856616e+121;
        bool r233092 = r233084 <= r233091;
        double r233093 = r233084 * r233084;
        double r233094 = t;
        double r233095 = a;
        double r233096 = r233094 * r233095;
        double r233097 = r233093 - r233096;
        double r233098 = sqrt(r233097);
        double r233099 = cbrt(r233098);
        double r233100 = r233099 * r233099;
        double r233101 = fabs(r233100);
        double r233102 = cbrt(r233084);
        double r233103 = r233102 * r233102;
        double r233104 = r233101 / r233103;
        double r233105 = r233088 / r233104;
        double r233106 = cbrt(r233097);
        double r233107 = sqrt(r233106);
        double r233108 = r233107 / r233102;
        double r233109 = r233087 / r233108;
        double r233110 = r233105 * r233109;
        double r233111 = r233092 ? r233110 : r233089;
        double r233112 = r233086 ? r233090 : r233111;
        return r233112;
}

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

Original24.3
Target7.4
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.7186746230671153e+135

    1. Initial program 50.6

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

    2. Taylor expanded around -inf 1.5

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

    3. Simplified1.5

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

    if -2.7186746230671153e+135 < z < 2.130897152856616e+121

    1. Initial program 10.5

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

    3. Applied associate-/l*8.7

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

    5. Applied add-cube-cbrt9.4

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

    6. Applied add-cube-cbrt9.4

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

    7. Applied sqrt-prod9.4

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

    8. Applied times-frac9.4

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

    9. Applied times-frac7.6

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

    10. Simplified7.6

      \[\leadsto \color{blue}{\frac{x}{\frac{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt7.6

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

    13. Applied cbrt-prod7.4

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

    if 2.130897152856616e+121 < z

    1. Initial program 46.6

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

    2. Taylor expanded around inf 2.0

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

    3. Simplified2.0

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.718674623067115332199814933152861240084 \cdot 10^{135}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \le 2.130897152856615796932912078487397342387 \cdot 10^{121}:\\ \;\;\;\;\frac{x}{\frac{\left|\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right|}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 -3.1921305903852764e46) (- (* y x)) (if (< z 5.9762681209208942e90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))