Average Error: 4.7 → 0.4
Time: 12.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.646424278606502678263957997837655215247 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.266911076549827575105216650109113541904 \cdot 10^{-282}:\\ \;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.632482779017283391313108001321509173255 \cdot 10^{260}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\
\;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.646424278606502678263957997837655215247 \cdot 10^{-233}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.266911076549827575105216650109113541904 \cdot 10^{-282}:\\
\;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.632482779017283391313108001321509173255 \cdot 10^{260}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r1606005 = x;
        double r1606006 = y;
        double r1606007 = z;
        double r1606008 = r1606006 / r1606007;
        double r1606009 = t;
        double r1606010 = 1.0;
        double r1606011 = r1606010 - r1606007;
        double r1606012 = r1606009 / r1606011;
        double r1606013 = r1606008 - r1606012;
        double r1606014 = r1606005 * r1606013;
        return r1606014;
}
↓
double f(double x, double y, double z, double t) {
        double r1606015 = y;
        double r1606016 = z;
        double r1606017 = r1606015 / r1606016;
        double r1606018 = t;
        double r1606019 = 1.0;
        double r1606020 = r1606019 - r1606016;
        double r1606021 = r1606018 / r1606020;
        double r1606022 = r1606017 - r1606021;
        double r1606023 = -inf.0;
        bool r1606024 = r1606022 <= r1606023;
        double r1606025 = x;
        double r1606026 = r1606015 * r1606020;
        double r1606027 = r1606016 * r1606018;
        double r1606028 = r1606026 - r1606027;
        double r1606029 = r1606025 * r1606028;
        double r1606030 = r1606016 * r1606020;
        double r1606031 = r1606029 / r1606030;
        double r1606032 = -3.6464242786065027e-233;
        bool r1606033 = r1606022 <= r1606032;
        double r1606034 = 1.0;
        double r1606035 = r1606020 / r1606018;
        double r1606036 = r1606034 / r1606035;
        double r1606037 = r1606017 - r1606036;
        double r1606038 = r1606025 * r1606037;
        double r1606039 = 1.2669110765498276e-282;
        bool r1606040 = r1606022 <= r1606039;
        double r1606041 = r1606019 / r1606016;
        double r1606042 = r1606041 + r1606034;
        double r1606043 = r1606018 * r1606025;
        double r1606044 = r1606043 / r1606016;
        double r1606045 = r1606042 * r1606044;
        double r1606046 = r1606025 * r1606015;
        double r1606047 = r1606046 / r1606016;
        double r1606048 = r1606045 + r1606047;
        double r1606049 = 3.6324827790172834e+260;
        bool r1606050 = r1606022 <= r1606049;
        double r1606051 = r1606050 ? r1606038 : r1606031;
        double r1606052 = r1606040 ? r1606048 : r1606051;
        double r1606053 = r1606033 ? r1606038 : r1606052;
        double r1606054 = r1606024 ? r1606031 : r1606053;
        return r1606054;
}

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

Original4.7
Target4.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 3.6324827790172834e+260 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 43.0

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

    3. Applied frac-sub44.0

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

    4. Applied associate-*r/1.3

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -3.6464242786065027e-233 or 1.2669110765498276e-282 < (- (/ y z) (/ t (- 1.0 z))) < 3.6324827790172834e+260

    1. Initial program 0.2

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

    3. Applied clear-num0.3

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

    if -3.6464242786065027e-233 < (- (/ y z) (/ t (- 1.0 z))) < 1.2669110765498276e-282

    1. Initial program 12.4

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -3.646424278606502678263957997837655215247 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.266911076549827575105216650109113541904 \cdot 10^{-282}:\\ \;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.632482779017283391313108001321509173255 \cdot 10^{260}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

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