Average Error: 4.8 → 1.8
Time: 29.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.815360187890604728685296818871955976429 \cdot 10^{158} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.87103258485720813958555940057997964078 \cdot 10^{143}\right):\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.815360187890604728685296818871955976429 \cdot 10^{158} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.87103258485720813958555940057997964078 \cdot 10^{143}\right):\\
\;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r211951 = x;
        double r211952 = y;
        double r211953 = z;
        double r211954 = r211952 / r211953;
        double r211955 = t;
        double r211956 = 1.0;
        double r211957 = r211956 - r211953;
        double r211958 = r211955 / r211957;
        double r211959 = r211954 - r211958;
        double r211960 = r211951 * r211959;
        return r211960;
}
↓
double f(double x, double y, double z, double t) {
        double r211961 = y;
        double r211962 = z;
        double r211963 = r211961 / r211962;
        double r211964 = t;
        double r211965 = 1.0;
        double r211966 = r211965 - r211962;
        double r211967 = r211964 / r211966;
        double r211968 = r211963 - r211967;
        double r211969 = -3.8153601878906047e+158;
        bool r211970 = r211968 <= r211969;
        double r211971 = 1.8710325848572081e+143;
        bool r211972 = r211968 <= r211971;
        double r211973 = !r211972;
        bool r211974 = r211970 || r211973;
        double r211975 = x;
        double r211976 = r211975 * r211961;
        double r211977 = r211976 / r211962;
        double r211978 = -r211967;
        double r211979 = r211975 * r211978;
        double r211980 = r211977 + r211979;
        double r211981 = 1.0;
        double r211982 = r211966 / r211964;
        double r211983 = r211981 / r211982;
        double r211984 = r211963 - r211983;
        double r211985 = r211975 * r211984;
        double r211986 = r211974 ? r211980 : r211985;
        return r211986;
}

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

Original4.8
Target4.5
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -3.8153601878906047e+158 or 1.8710325848572081e+143 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 14.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg14.0

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

    4. Applied distribute-lft-in14.0

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

    5. Simplified1.7

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

    if -3.8153601878906047e+158 < (- (/ y z) (/ t (- 1.0 z))) < 1.8710325848572081e+143

    1. Initial program 1.7

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied clear-num1.8

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.815360187890604728685296818871955976429 \cdot 10^{158} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.87103258485720813958555940057997964078 \cdot 10^{143}\right):\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

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