Average Error: 2.0 → 1.7
Time: 2.9s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.50347376632767726 \cdot 10^{-49} \lor \neg \left(y \le 3.8833316150033781 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{y}{\left(z - t\right) \cdot x}} + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.50347376632767726 \cdot 10^{-49} \lor \neg \left(y \le 3.8833316150033781 \cdot 10^{-63}\right):\\
\;\;\;\;x \cdot \frac{z - t}{y} + t\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r461941 = x;
        double r461942 = y;
        double r461943 = r461941 / r461942;
        double r461944 = z;
        double r461945 = t;
        double r461946 = r461944 - r461945;
        double r461947 = r461943 * r461946;
        double r461948 = r461947 + r461945;
        return r461948;
}
↓
double f(double x, double y, double z, double t) {
        double r461949 = y;
        double r461950 = -1.5034737663276773e-49;
        bool r461951 = r461949 <= r461950;
        double r461952 = 3.883331615003378e-63;
        bool r461953 = r461949 <= r461952;
        double r461954 = !r461953;
        bool r461955 = r461951 || r461954;
        double r461956 = x;
        double r461957 = z;
        double r461958 = t;
        double r461959 = r461957 - r461958;
        double r461960 = r461959 / r461949;
        double r461961 = r461956 * r461960;
        double r461962 = r461961 + r461958;
        double r461963 = 1.0;
        double r461964 = r461959 * r461956;
        double r461965 = r461949 / r461964;
        double r461966 = r461963 / r461965;
        double r461967 = r461963 * r461966;
        double r461968 = r461967 + r461958;
        double r461969 = r461955 ? r461962 : r461968;
        return r461969;
}

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

Original2.0
Target2.3
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.5034737663276773e-49 or 3.883331615003378e-63 < y

    1. Initial program 1.1

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

    3. Applied div-inv1.1

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

    4. Applied associate-*l*1.4

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

    5. Simplified1.4

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

    if -1.5034737663276773e-49 < y < 3.883331615003378e-63

    1. Initial program 4.4

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

    3. Applied *-un-lft-identity4.4

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

    4. Applied *-un-lft-identity4.4

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

    5. Applied times-frac4.4

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

    6. Applied associate-*l*4.4

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

    7. Simplified2.3

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

    9. Applied clear-num2.3

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{1}{\frac{y}{\left(z - t\right) \cdot x}}} + t\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.50347376632767726 \cdot 10^{-49} \lor \neg \left(y \le 3.8833316150033781 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{y}{\left(z - t\right) \cdot x}} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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