Average Error: 7.2 → 3.0
Time: 5.0s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.412584965026174356547481107780761150371 \cdot 10^{211}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -2.580684021360039689076100657919132626783 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{y}{t \cdot z - x} \cdot z + x}{\left(x + 1\right) \cdot 1} - \sqrt[3]{{\left(\frac{\frac{x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \mathbf{elif}\;z \le -2.966830251509030236990636984311025104722 \cdot 10^{118}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 1.03829628730808273860363989281490107116 \cdot 10^{46}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t \cdot z - x} \cdot z + x}{\left(x + 1\right) \cdot 1} - \sqrt[3]{{\left(\frac{\frac{x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.412584965026174356547481107780761150371 \cdot 10^{211}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{elif}\;z \le -2.580684021360039689076100657919132626783 \cdot 10^{160}:\\
\;\;\;\;\frac{\frac{y}{t \cdot z - x} \cdot z + x}{\left(x + 1\right) \cdot 1} - \sqrt[3]{{\left(\frac{\frac{x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\

\mathbf{elif}\;z \le -2.966830251509030236990636984311025104722 \cdot 10^{118}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{elif}\;z \le 1.03829628730808273860363989281490107116 \cdot 10^{46}:\\
\;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r591060 = x;
        double r591061 = y;
        double r591062 = z;
        double r591063 = r591061 * r591062;
        double r591064 = r591063 - r591060;
        double r591065 = t;
        double r591066 = r591065 * r591062;
        double r591067 = r591066 - r591060;
        double r591068 = r591064 / r591067;
        double r591069 = r591060 + r591068;
        double r591070 = 1.0;
        double r591071 = r591060 + r591070;
        double r591072 = r591069 / r591071;
        return r591072;
}
↓
double f(double x, double y, double z, double t) {
        double r591073 = z;
        double r591074 = -1.4125849650261744e+211;
        bool r591075 = r591073 <= r591074;
        double r591076 = x;
        double r591077 = y;
        double r591078 = t;
        double r591079 = r591077 / r591078;
        double r591080 = r591076 + r591079;
        double r591081 = 1.0;
        double r591082 = r591076 + r591081;
        double r591083 = r591080 / r591082;
        double r591084 = -2.5806840213600397e+160;
        bool r591085 = r591073 <= r591084;
        double r591086 = r591078 * r591073;
        double r591087 = r591086 - r591076;
        double r591088 = r591077 / r591087;
        double r591089 = r591088 * r591073;
        double r591090 = r591089 + r591076;
        double r591091 = 1.0;
        double r591092 = r591082 * r591091;
        double r591093 = r591090 / r591092;
        double r591094 = r591076 / r591087;
        double r591095 = r591094 / r591082;
        double r591096 = 3.0;
        double r591097 = pow(r591095, r591096);
        double r591098 = cbrt(r591097);
        double r591099 = r591093 - r591098;
        double r591100 = -2.9668302515090302e+118;
        bool r591101 = r591073 <= r591100;
        double r591102 = 1.0382962873080827e+46;
        bool r591103 = r591073 <= r591102;
        double r591104 = r591077 * r591073;
        double r591105 = r591104 - r591076;
        double r591106 = r591105 / r591087;
        double r591107 = r591076 + r591106;
        double r591108 = r591091 / r591082;
        double r591109 = r591107 * r591108;
        double r591110 = r591103 ? r591109 : r591099;
        double r591111 = r591101 ? r591083 : r591110;
        double r591112 = r591085 ? r591099 : r591111;
        double r591113 = r591075 ? r591083 : r591112;
        return r591113;
}

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

Original7.2
Target0.4
Herbie3.0
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.4125849650261744e+211 or -2.5806840213600397e+160 < z < -2.9668302515090302e+118

    1. Initial program 21.8

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

    2. Taylor expanded around inf 7.5

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

    if -1.4125849650261744e+211 < z < -2.5806840213600397e+160 or 1.0382962873080827e+46 < z

    1. Initial program 17.5

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

    3. Applied div-sub17.5

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

    4. Applied associate-+r-17.5

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

    5. Applied div-sub17.5

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

    6. Simplified6.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
    7. Using strategy rm

    8. Applied fma-udef6.3

      \[\leadsto \frac{\color{blue}{\frac{y}{t \cdot z - x} \cdot z + x}}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube6.3

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

    11. Applied add-cbrt-cube7.2

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

    12. Applied add-cbrt-cube30.3

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

    13. Applied cbrt-undiv30.3

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

    14. Applied cbrt-undiv30.3

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

    15. Simplified6.3

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

    if -2.9668302515090302e+118 < z < 1.0382962873080827e+46

    1. Initial program 1.0

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

    3. Applied div-inv1.1

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.412584965026174356547481107780761150371 \cdot 10^{211}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -2.580684021360039689076100657919132626783 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{y}{t \cdot z - x} \cdot z + x}{\left(x + 1\right) \cdot 1} - \sqrt[3]{{\left(\frac{\frac{x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \mathbf{elif}\;z \le -2.966830251509030236990636984311025104722 \cdot 10^{118}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 1.03829628730808273860363989281490107116 \cdot 10^{46}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t \cdot z - x} \cdot z + x}{\left(x + 1\right) \cdot 1} - \sqrt[3]{{\left(\frac{\frac{x}{t \cdot z - x}}{x + 1}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))