Average Error: 4.8 → 3.1
Time: 5.9s
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 r375060 = x;
        double r375061 = y;
        double r375062 = z;
        double r375063 = r375061 / r375062;
        double r375064 = t;
        double r375065 = 1.0;
        double r375066 = r375065 - r375062;
        double r375067 = r375064 / r375066;
        double r375068 = r375063 - r375067;
        double r375069 = r375060 * r375068;
        return r375069;
}
↓
double f(double x, double y, double z, double t) {
        double r375070 = y;
        double r375071 = z;
        double r375072 = r375070 / r375071;
        double r375073 = t;
        double r375074 = 1.0;
        double r375075 = r375074 - r375071;
        double r375076 = r375073 / r375075;
        double r375077 = r375072 - r375076;
        double r375078 = 5.623781674200428e+272;
        bool r375079 = r375077 <= r375078;
        double r375080 = x;
        double r375081 = 1.0;
        double r375082 = r375080 / r375081;
        double r375083 = -r375076;
        double r375084 = r375080 * r375083;
        double r375085 = fma(r375082, r375072, r375084);
        double r375086 = r375080 * r375070;
        double r375087 = r375071 / r375086;
        double r375088 = r375081 / r375087;
        double r375089 = r375088 + r375084;
        double r375090 = r375079 ? r375085 : r375089;
        return r375090;
}

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)))))