Average Error: 4.8 → 0.4
Time: 10.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.2592621423955549 \cdot 10^{306}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.9384031107510682 \cdot 10^{-253}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.2094051295440155 \cdot 10^{253}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.2592621423955549 \cdot 10^{306}:\\
\;\;\;\;\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 -6.9384031107510682 \cdot 10^{-253}:\\
\;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.2094051295440155 \cdot 10^{253}:\\
\;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\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 r387010 = x;
        double r387011 = y;
        double r387012 = z;
        double r387013 = r387011 / r387012;
        double r387014 = t;
        double r387015 = 1.0;
        double r387016 = r387015 - r387012;
        double r387017 = r387014 / r387016;
        double r387018 = r387013 - r387017;
        double r387019 = r387010 * r387018;
        return r387019;
}
↓
double f(double x, double y, double z, double t) {
        double r387020 = y;
        double r387021 = z;
        double r387022 = r387020 / r387021;
        double r387023 = t;
        double r387024 = 1.0;
        double r387025 = r387024 - r387021;
        double r387026 = r387023 / r387025;
        double r387027 = r387022 - r387026;
        double r387028 = -2.259262142395555e+306;
        bool r387029 = r387027 <= r387028;
        double r387030 = x;
        double r387031 = r387020 * r387025;
        double r387032 = r387021 * r387023;
        double r387033 = r387031 - r387032;
        double r387034 = r387030 * r387033;
        double r387035 = r387021 * r387025;
        double r387036 = r387034 / r387035;
        double r387037 = -6.938403110751068e-253;
        bool r387038 = r387027 <= r387037;
        double r387039 = 1.0;
        double r387040 = r387039 / r387021;
        double r387041 = r387039 / r387025;
        double r387042 = r387023 * r387041;
        double r387043 = -r387042;
        double r387044 = fma(r387020, r387040, r387043);
        double r387045 = -r387023;
        double r387046 = fma(r387045, r387041, r387042);
        double r387047 = r387044 + r387046;
        double r387048 = r387030 * r387047;
        double r387049 = -0.0;
        bool r387050 = r387027 <= r387049;
        double r387051 = r387030 / r387021;
        double r387052 = r387023 * r387030;
        double r387053 = 2.0;
        double r387054 = pow(r387021, r387053);
        double r387055 = r387052 / r387054;
        double r387056 = r387052 / r387021;
        double r387057 = fma(r387024, r387055, r387056);
        double r387058 = fma(r387051, r387020, r387057);
        double r387059 = 6.209405129544016e+253;
        bool r387060 = r387027 <= r387059;
        double r387061 = r387060 ? r387048 : r387036;
        double r387062 = r387050 ? r387058 : r387061;
        double r387063 = r387038 ? r387048 : r387062;
        double r387064 = r387029 ? r387036 : r387063;
        return r387064;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.8
Target4.3
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -2.259262142395555e+306 or 6.209405129544016e+253 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 43.9

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

    3. Applied frac-sub44.9

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

    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 -2.259262142395555e+306 < (- (/ y z) (/ t (- 1.0 z))) < -6.938403110751068e-253 or -0.0 < (- (/ y z) (/ t (- 1.0 z))) < 6.209405129544016e+253

    1. Initial program 0.3

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

    3. Applied clear-num0.4

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

    5. Applied add-cube-cbrt0.8

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

    6. Applied div-inv0.9

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

    7. Applied prod-diff0.9

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

    8. Simplified0.3

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

    9. Simplified0.3

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

    if -6.938403110751068e-253 < (- (/ y z) (/ t (- 1.0 z))) < -0.0

    1. Initial program 16.0

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

    3. Applied clear-num16.4

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

    5. Applied add-cube-cbrt16.4

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

    6. Applied div-inv16.4

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

    7. Applied prod-diff16.4

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

    8. Simplified16.0

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

    9. Simplified16.0

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

    10. Taylor expanded around inf 0.1

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

    11. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.2592621423955549 \cdot 10^{306}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.9384031107510682 \cdot 10^{-253}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.2094051295440155 \cdot 10^{253}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, \frac{1}{z}, -t \cdot \frac{1}{1 - z}\right) + \mathsf{fma}\left(-t, \frac{1}{1 - z}, t \cdot \frac{1}{1 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))))