Average Error: 4.6 → 1.3
Time: 18.7s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -5.348998325828909283670937800087002307001 \cdot 10^{228}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.298652415712523287983693451925423845018 \cdot 10^{-225}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 9.455037476058961105673774937517387989227 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right) + x \cdot \left(\frac{-t}{1 - z} + \frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.370485689795873119951086996921135257728 \cdot 10^{233}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -5.348998325828909283670937800087002307001 \cdot 10^{228}:\\
\;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.298652415712523287983693451925423845018 \cdot 10^{-225}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 9.455037476058961105673774937517387989227 \cdot 10^{-126}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right) + x \cdot \left(\frac{-t}{1 - z} + \frac{t}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.370485689795873119951086996921135257728 \cdot 10^{233}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r321185 = x;
        double r321186 = y;
        double r321187 = z;
        double r321188 = r321186 / r321187;
        double r321189 = t;
        double r321190 = 1.0;
        double r321191 = r321190 - r321187;
        double r321192 = r321189 / r321191;
        double r321193 = r321188 - r321192;
        double r321194 = r321185 * r321193;
        return r321194;
}
↓
double f(double x, double y, double z, double t) {
        double r321195 = y;
        double r321196 = z;
        double r321197 = r321195 / r321196;
        double r321198 = t;
        double r321199 = 1.0;
        double r321200 = r321199 - r321196;
        double r321201 = r321198 / r321200;
        double r321202 = r321197 - r321201;
        double r321203 = -5.348998325828909e+228;
        bool r321204 = r321202 <= r321203;
        double r321205 = x;
        double r321206 = r321205 * r321195;
        double r321207 = r321206 / r321196;
        double r321208 = -r321201;
        double r321209 = r321205 * r321208;
        double r321210 = r321207 + r321209;
        double r321211 = -2.2986524157125233e-225;
        bool r321212 = r321202 <= r321211;
        double r321213 = 1.0;
        double r321214 = r321200 / r321198;
        double r321215 = r321213 / r321214;
        double r321216 = r321197 - r321215;
        double r321217 = r321205 * r321216;
        double r321218 = 9.455037476058961e-126;
        bool r321219 = r321202 <= r321218;
        double r321220 = r321199 / r321196;
        double r321221 = r321220 + r321213;
        double r321222 = r321198 * r321205;
        double r321223 = r321222 / r321196;
        double r321224 = r321221 * r321223;
        double r321225 = fma(r321197, r321205, r321224);
        double r321226 = -r321198;
        double r321227 = r321226 / r321200;
        double r321228 = r321227 + r321201;
        double r321229 = r321205 * r321228;
        double r321230 = r321225 + r321229;
        double r321231 = 1.3704856897958731e+233;
        bool r321232 = r321202 <= r321231;
        double r321233 = r321232 ? r321217 : r321210;
        double r321234 = r321219 ? r321230 : r321233;
        double r321235 = r321212 ? r321217 : r321234;
        double r321236 = r321204 ? r321210 : r321235;
        return r321236;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target4.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -5.348998325828909e+228 or 1.3704856897958731e+233 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 26.1

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

    3. Applied sub-neg26.1

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

    4. Applied distribute-lft-in26.1

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

    5. Simplified0.5

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

    if -5.348998325828909e+228 < (- (/ y z) (/ t (- 1.0 z))) < -2.2986524157125233e-225 or 9.455037476058961e-126 < (- (/ y z) (/ t (- 1.0 z))) < 1.3704856897958731e+233

    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 -2.2986524157125233e-225 < (- (/ y z) (/ t (- 1.0 z))) < 9.455037476058961e-126

    1. Initial program 6.5

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

    3. Applied clear-num7.0

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

    5. Applied add-cube-cbrt7.3

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}}\right)\]

    6. Applied add-cube-cbrt7.5

      \[\leadsto x \cdot \left(\frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} - \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\]

    7. Applied add-cube-cbrt7.6

      \[\leadsto x \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} - \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\]

    8. Applied times-frac7.6

      \[\leadsto x \cdot \left(\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}} - \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\]

    9. Applied prod-diff7.6

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{y}}{\sqrt[3]{z}}, -\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\right)\right)}\]

    10. Applied distribute-lft-in7.6

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{y}}{\sqrt[3]{z}}, -\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\right) + x \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\right)}\]

    11. Simplified6.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{y}}{\sqrt[3]{z}}, \frac{-t}{1 - z}\right) \cdot x} + x \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}, \sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{1 - z}{t}}} \cdot \sqrt[3]{\frac{1}{\frac{1 - z}{t}}}\right)\right)\]

    12. Simplified6.9

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{y}}{\sqrt[3]{z}}, \frac{-t}{1 - z}\right) \cdot x + \color{blue}{x \cdot \left(\frac{-t}{1 - z} + \frac{t}{1 - z}\right)}\]

    13. Taylor expanded around inf 46.7

      \[\leadsto \color{blue}{\left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \left(\frac{t \cdot x}{z} + e^{\frac{1}{3} \cdot \left(2 \cdot \log \left(\frac{1}{z}\right) - 2 \cdot \log \left(\frac{1}{y}\right)\right)} \cdot \left(e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{z}\right) - \log \left(\frac{1}{y}\right)\right)} \cdot x\right)\right)\right)} + x \cdot \left(\frac{-t}{1 - z} + \frac{t}{1 - z}\right)\]

    14. Simplified6.1

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -5.348998325828909283670937800087002307001 \cdot 10^{228}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.298652415712523287983693451925423845018 \cdot 10^{-225}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 9.455037476058961105673774937517387989227 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z}\right) + x \cdot \left(\frac{-t}{1 - z} + \frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.370485689795873119951086996921135257728 \cdot 10^{233}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))))