Average Error: 25.1 → 6.1
Time: 9.3s
Precision: 64
\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\
\;\;\;\;x \cdot \left(y \cdot \frac{z}{\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 r555103 = x;
        double r555104 = y;
        double r555105 = r555103 * r555104;
        double r555106 = z;
        double r555107 = r555105 * r555106;
        double r555108 = r555106 * r555106;
        double r555109 = t;
        double r555110 = a;
        double r555111 = r555109 * r555110;
        double r555112 = r555108 - r555111;
        double r555113 = sqrt(r555112);
        double r555114 = r555107 / r555113;
        return r555114;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r555115 = z;
        double r555116 = -9.341949798077739e+151;
        bool r555117 = r555115 <= r555116;
        double r555118 = x;
        double r555119 = y;
        double r555120 = -r555119;
        double r555121 = r555118 * r555120;
        double r555122 = 2.2349389384476415e+125;
        bool r555123 = r555115 <= r555122;
        double r555124 = r555115 * r555115;
        double r555125 = t;
        double r555126 = a;
        double r555127 = r555125 * r555126;
        double r555128 = r555124 - r555127;
        double r555129 = sqrt(r555128);
        double r555130 = r555115 / r555129;
        double r555131 = r555119 * r555130;
        double r555132 = r555118 * r555131;
        double r555133 = r555118 * r555119;
        double r555134 = r555123 ? r555132 : r555133;
        double r555135 = r555117 ? r555121 : r555134;
        return r555135;
}

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

Original25.1
Target7.5
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -9.341949798077739e+151

    1. Initial program 53.6

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

    3. Applied *-un-lft-identity53.6

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

    4. Applied sqrt-prod53.6

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

    5. Applied times-frac53.2

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

    6. Simplified53.2

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

    8. Applied associate-*l*53.2

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

    9. Taylor expanded around -inf 1.5

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

    10. Simplified1.5

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

    if -9.341949798077739e+151 < z < 2.2349389384476415e+125

    1. Initial program 11.5

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

    3. Applied *-un-lft-identity11.5

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

    4. Applied sqrt-prod11.5

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

    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 associate-*l*8.6

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

    if 2.2349389384476415e+125 < z

    1. Initial program 47.2

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.34194979807773912 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 2.2349389384476415 \cdot 10^{125}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))