Average Error: 7.5 → 2.3
Time: 10.8s
Precision: 64
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 4.0510841144545337 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\left(t - z\right) \cdot \frac{y - z}{x}}\\ \mathbf{elif}\;z \le 1.5692981510658075 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{y - z} \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \end{array}\]
\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le 4.0510841144545337 \cdot 10^{-255}:\\
\;\;\;\;\frac{1}{\left(t - z\right) \cdot \frac{y - z}{x}}\\

\mathbf{elif}\;z \le 1.5692981510658075 \cdot 10^{-125}:\\
\;\;\;\;\frac{1}{y - z} \cdot \frac{x}{t - z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r942109 = x;
        double r942110 = y;
        double r942111 = z;
        double r942112 = r942110 - r942111;
        double r942113 = t;
        double r942114 = r942113 - r942111;
        double r942115 = r942112 * r942114;
        double r942116 = r942109 / r942115;
        return r942116;
}
↓
double f(double x, double y, double z, double t) {
        double r942117 = z;
        double r942118 = 4.0510841144545337e-255;
        bool r942119 = r942117 <= r942118;
        double r942120 = 1.0;
        double r942121 = t;
        double r942122 = r942121 - r942117;
        double r942123 = y;
        double r942124 = r942123 - r942117;
        double r942125 = x;
        double r942126 = r942124 / r942125;
        double r942127 = r942122 * r942126;
        double r942128 = r942120 / r942127;
        double r942129 = 1.5692981510658075e-125;
        bool r942130 = r942117 <= r942129;
        double r942131 = r942120 / r942124;
        double r942132 = r942125 / r942122;
        double r942133 = r942131 * r942132;
        double r942134 = r942125 / r942124;
        double r942135 = r942134 / r942122;
        double r942136 = r942130 ? r942133 : r942135;
        double r942137 = r942119 ? r942128 : r942136;
        return r942137;
}

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
Target8.3
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \lt 0.0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 4.0510841144545337e-255

    1. Initial program 7.5

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

    3. Applied associate-/r*2.5

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

    5. Applied clear-num2.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y - z}{x}}}}{t - z}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity2.8

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

    8. Applied *-un-lft-identity2.8

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

    9. Applied times-frac2.8

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

    10. Applied *-un-lft-identity2.8

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

    11. Applied times-frac2.8

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

    12. Applied associate-/l*3.3

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

    13. Simplified3.0

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

    if 4.0510841144545337e-255 < z < 1.5692981510658075e-125

    1. Initial program 5.3

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

    3. Applied *-un-lft-identity5.3

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

    4. Applied times-frac3.9

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

    if 1.5692981510658075e-125 < z

    1. Initial program 8.1

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

    3. Applied associate-/r*0.7

      \[\leadsto \color{blue}{\frac{\frac{x}{y - z}}{t - z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 4.0510841144545337 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\left(t - z\right) \cdot \frac{y - z}{x}}\\ \mathbf{elif}\;z \le 1.5692981510658075 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{y - z} \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1 (* (- y z) (- t z)))))

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