Average Error: 4.8 → 3.1
Time: 5.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.623781674200427894596744847477299567005 \cdot 10^{272}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.623781674200427894596744847477299567005 \cdot 10^{272}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r335833 = x;
        double r335834 = y;
        double r335835 = z;
        double r335836 = r335834 / r335835;
        double r335837 = t;
        double r335838 = 1.0;
        double r335839 = r335838 - r335835;
        double r335840 = r335837 / r335839;
        double r335841 = r335836 - r335840;
        double r335842 = r335833 * r335841;
        return r335842;
}
↓
double f(double x, double y, double z, double t) {
        double r335843 = y;
        double r335844 = z;
        double r335845 = r335843 / r335844;
        double r335846 = t;
        double r335847 = 1.0;
        double r335848 = r335847 - r335844;
        double r335849 = r335846 / r335848;
        double r335850 = r335845 - r335849;
        double r335851 = 5.623781674200428e+272;
        bool r335852 = r335850 <= r335851;
        double r335853 = x;
        double r335854 = 1.0;
        double r335855 = r335853 / r335854;
        double r335856 = -r335849;
        double r335857 = r335853 * r335856;
        double r335858 = fma(r335855, r335845, r335857);
        double r335859 = r335853 * r335843;
        double r335860 = r335844 / r335859;
        double r335861 = r335854 / r335860;
        double r335862 = r335861 + r335857;
        double r335863 = r335852 ? r335858 : r335862;
        return r335863;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.8
Target4.4
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < 5.623781674200428e+272

    1. Initial program 3.2

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

    3. Applied div-inv3.3

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

    4. Applied fma-neg3.3

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

    6. Applied fma-udef3.3

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

    7. Applied distribute-lft-in3.3

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

    8. Simplified5.9

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

    10. Applied *-un-lft-identity5.9

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

    11. Applied times-frac3.2

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

    12. Applied fma-def3.2

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

    if 5.623781674200428e+272 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 38.3

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

    3. Applied div-inv38.4

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

    4. Applied fma-neg38.4

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

    6. Applied fma-udef38.4

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

    7. Applied distribute-lft-in38.4

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

    8. Simplified0.2

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

    10. Applied clear-num0.3

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.623781674200427894596744847477299567005 \cdot 10^{272}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))))