Average Error: 7.5 → 4.7
Time: 4.9s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.961108853191947737319683888969508909152 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 751958545345597:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.906429000813494708008533930580696575797 \cdot 10^{305}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\
\;\;\;\;\sqrt[3]{1}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.961108853191947737319683888969508909152 \cdot 10^{-21}:\\
\;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 751958545345597:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.906429000813494708008533930580696575797 \cdot 10^{305}:\\
\;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r715709 = x;
        double r715710 = y;
        double r715711 = z;
        double r715712 = r715710 * r715711;
        double r715713 = r715712 - r715709;
        double r715714 = t;
        double r715715 = r715714 * r715711;
        double r715716 = r715715 - r715709;
        double r715717 = r715713 / r715716;
        double r715718 = r715709 + r715717;
        double r715719 = 1.0;
        double r715720 = r715709 + r715719;
        double r715721 = r715718 / r715720;
        return r715721;
}
↓
double f(double x, double y, double z, double t) {
        double r715722 = x;
        double r715723 = y;
        double r715724 = z;
        double r715725 = r715723 * r715724;
        double r715726 = r715725 - r715722;
        double r715727 = t;
        double r715728 = r715727 * r715724;
        double r715729 = r715728 - r715722;
        double r715730 = r715726 / r715729;
        double r715731 = r715722 + r715730;
        double r715732 = 1.0;
        double r715733 = r715722 + r715732;
        double r715734 = r715731 / r715733;
        double r715735 = -inf.0;
        bool r715736 = r715734 <= r715735;
        double r715737 = 1.0;
        double r715738 = cbrt(r715737);
        double r715739 = 5.961108853191948e-21;
        bool r715740 = r715734 <= r715739;
        double r715741 = sqrt(r715733);
        double r715742 = r715737 / r715741;
        double r715743 = r715731 / r715741;
        double r715744 = r715742 * r715743;
        double r715745 = 751958545345597.0;
        bool r715746 = r715734 <= r715745;
        double r715747 = 3.0;
        double r715748 = pow(r715734, r715747);
        double r715749 = cbrt(r715748);
        double r715750 = 3.906429000813495e+305;
        bool r715751 = r715734 <= r715750;
        double r715752 = r715723 / r715727;
        double r715753 = r715722 + r715752;
        double r715754 = r715722 * r715722;
        double r715755 = r715732 * r715732;
        double r715756 = r715754 - r715755;
        double r715757 = r715753 / r715756;
        double r715758 = r715722 - r715732;
        double r715759 = r715757 * r715758;
        double r715760 = r715751 ? r715744 : r715759;
        double r715761 = r715746 ? r715749 : r715760;
        double r715762 = r715740 ? r715744 : r715761;
        double r715763 = r715736 ? r715738 : r715762;
        return r715763;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.5
Target0.3
Herbie4.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

    1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube64.0

      \[\leadsto \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\color{blue}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\]

    4. Applied add-cbrt-cube64.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)}}}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}\]

    5. Applied cbrt-undiv64.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\]

    6. Simplified64.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\right)}^{3}}}\]

    7. Taylor expanded around inf 44.5

      \[\leadsto \sqrt[3]{\color{blue}{1}}\]

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 5.961108853191948e-21 or 751958545345597.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 3.906429000813495e+305

    1. Initial program 1.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt3.7

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

    4. Applied *-un-lft-identity3.7

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

    5. Applied times-frac3.7

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

    if 5.961108853191948e-21 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 751958545345597.0

    1. Initial program 0.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube34.9

      \[\leadsto \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\color{blue}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\]

    4. Applied add-cbrt-cube35.9

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)}}}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}\]

    5. Applied cbrt-undiv35.9

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)\right) \cdot \left(x + \frac{y \cdot z - x}{t \cdot z - x}\right)}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\]

    6. Simplified0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\right)}^{3}}}\]

    if 3.906429000813495e+305 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 63.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    2. Taylor expanded around inf 10.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
    3. Using strategy rm

    4. Applied flip-+23.9

      \[\leadsto \frac{x + \frac{y}{t}}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}}\]

    5. Applied associate-/r/23.9

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.961108853191947737319683888969508909152 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 751958545345597:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.906429000813494708008533930580696575797 \cdot 10^{305}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1}} \cdot \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))