Average Error: 4.5 → 0.5
Time: 16.4s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.145223723342324046712267811210494216144 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.051161075875252365315860485678053244137 \cdot 10^{156}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{t}{1 - z} \cdot \left(-x\right)\\ \end{array}\]
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 -2.145223723342324046712267811210494216144 \cdot 10^{-232}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.051161075875252365315860485678053244137 \cdot 10^{156}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r571955 = x;
        double r571956 = y;
        double r571957 = z;
        double r571958 = r571956 / r571957;
        double r571959 = t;
        double r571960 = 1.0;
        double r571961 = r571960 - r571957;
        double r571962 = r571959 / r571961;
        double r571963 = r571958 - r571962;
        double r571964 = r571955 * r571963;
        return r571964;
}
↓
double f(double x, double y, double z, double t) {
        double r571965 = y;
        double r571966 = z;
        double r571967 = r571965 / r571966;
        double r571968 = t;
        double r571969 = 1.0;
        double r571970 = r571969 - r571966;
        double r571971 = r571968 / r571970;
        double r571972 = r571967 - r571971;
        double r571973 = -inf.0;
        bool r571974 = r571972 <= r571973;
        double r571975 = x;
        double r571976 = r571965 * r571970;
        double r571977 = r571966 * r571968;
        double r571978 = r571976 - r571977;
        double r571979 = r571975 * r571978;
        double r571980 = r571966 * r571970;
        double r571981 = r571979 / r571980;
        double r571982 = -2.145223723342324e-232;
        bool r571983 = r571972 <= r571982;
        double r571984 = 1.0;
        double r571985 = r571984 / r571970;
        double r571986 = r571968 * r571985;
        double r571987 = r571967 - r571986;
        double r571988 = r571975 * r571987;
        double r571989 = -0.0;
        bool r571990 = r571972 <= r571989;
        double r571991 = r571975 / r571966;
        double r571992 = r571969 / r571966;
        double r571993 = r571992 + r571984;
        double r571994 = r571968 * r571975;
        double r571995 = r571994 / r571966;
        double r571996 = r571993 * r571995;
        double r571997 = fma(r571991, r571965, r571996);
        double r571998 = 1.0511610758752524e+156;
        bool r571999 = r571972 <= r571998;
        double r572000 = r571975 * r571965;
        double r572001 = r572000 / r571966;
        double r572002 = -r571975;
        double r572003 = r571971 * r572002;
        double r572004 = r572001 + r572003;
        double r572005 = r571999 ? r571988 : r572004;
        double r572006 = r571990 ? r571997 : r572005;
        double r572007 = r571983 ? r571988 : r572006;
        double r572008 = r571974 ? r571981 : r572007;
        return r572008;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.5
Target4.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0

    1. Initial program 64.0

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

    3. Applied frac-sub64.0

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

    4. Applied associate-*r/0.3

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -2.145223723342324e-232 or -0.0 < (- (/ y z) (/ t (- 1.0 z))) < 1.0511610758752524e+156

    1. Initial program 1.2

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

    3. Applied div-inv1.2

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

    if -2.145223723342324e-232 < (- (/ y z) (/ t (- 1.0 z))) < -0.0

    1. Initial program 5.5

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

    3. Applied div-inv5.5

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

    4. Taylor expanded around inf 0.8

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

    5. Simplified0.5

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

    if 1.0511610758752524e+156 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 13.2

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

    3. Applied div-inv13.2

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

    5. Applied div-inv13.3

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

    6. Applied fma-neg13.2

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

    7. Simplified13.2

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

    9. Applied clear-num13.3

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

    11. Applied fma-udef13.3

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

    12. Applied distribute-lft-in13.3

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

    13. Simplified1.7

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

    14. Simplified1.7

      \[\leadsto \frac{x \cdot y}{z} + \color{blue}{\frac{t}{1 - z} \cdot \left(-x\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.145223723342324046712267811210494216144 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.051161075875252365315860485678053244137 \cdot 10^{156}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{t}{1 - z} \cdot \left(-x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))) -7.62322630331204244e-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)))))