Average Error: 4.8 → 0.9
Time: 16.2s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.35613759911877656476275151503320492094 \cdot 10^{280}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -5.000777057502044048634678543233616265407 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.961295342052216983850119223211706034603 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.387575495283554302798986221595555673559 \cdot 10^{280}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.35613759911877656476275151503320492094 \cdot 10^{280}:\\
\;\;\;\;\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 -5.000777057502044048634678543233616265407 \cdot 10^{-118}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.961295342052216983850119223211706034603 \cdot 10^{-199}:\\
\;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.387575495283554302798986221595555673559 \cdot 10^{280}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\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 r295677 = x;
        double r295678 = y;
        double r295679 = z;
        double r295680 = r295678 / r295679;
        double r295681 = t;
        double r295682 = 1.0;
        double r295683 = r295682 - r295679;
        double r295684 = r295681 / r295683;
        double r295685 = r295680 - r295684;
        double r295686 = r295677 * r295685;
        return r295686;
}
↓
double f(double x, double y, double z, double t) {
        double r295687 = y;
        double r295688 = z;
        double r295689 = r295687 / r295688;
        double r295690 = t;
        double r295691 = 1.0;
        double r295692 = r295691 - r295688;
        double r295693 = r295690 / r295692;
        double r295694 = r295689 - r295693;
        double r295695 = -1.3561375991187766e+280;
        bool r295696 = r295694 <= r295695;
        double r295697 = x;
        double r295698 = r295687 * r295692;
        double r295699 = r295688 * r295690;
        double r295700 = r295698 - r295699;
        double r295701 = r295697 * r295700;
        double r295702 = r295688 * r295692;
        double r295703 = r295701 / r295702;
        double r295704 = -5.000777057502044e-118;
        bool r295705 = r295694 <= r295704;
        double r295706 = 1.0;
        double r295707 = r295692 / r295690;
        double r295708 = r295706 / r295707;
        double r295709 = r295689 - r295708;
        double r295710 = r295697 * r295709;
        double r295711 = 5.961295342052217e-199;
        bool r295712 = r295694 <= r295711;
        double r295713 = r295691 / r295688;
        double r295714 = r295713 + r295706;
        double r295715 = r295690 * r295697;
        double r295716 = r295715 / r295688;
        double r295717 = r295714 * r295716;
        double r295718 = r295697 * r295687;
        double r295719 = r295718 / r295688;
        double r295720 = r295717 + r295719;
        double r295721 = 1.3875754952835543e+280;
        bool r295722 = r295694 <= r295721;
        double r295723 = r295722 ? r295710 : r295703;
        double r295724 = r295712 ? r295720 : r295723;
        double r295725 = r295705 ? r295710 : r295724;
        double r295726 = r295696 ? r295703 : r295725;
        return r295726;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.8
Target4.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -1.3561375991187766e+280 or 1.3875754952835543e+280 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 42.0

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

    3. Applied frac-sub42.7

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

    4. Applied associate-*r/0.9

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

    if -1.3561375991187766e+280 < (- (/ y z) (/ t (- 1.0 z))) < -5.000777057502044e-118 or 5.961295342052217e-199 < (- (/ y z) (/ t (- 1.0 z))) < 1.3875754952835543e+280

    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 -5.000777057502044e-118 < (- (/ y z) (/ t (- 1.0 z))) < 5.961295342052217e-199

    1. Initial program 6.2

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

    2. Taylor expanded around inf 3.4

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

    3. Simplified3.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.35613759911877656476275151503320492094 \cdot 10^{280}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -5.000777057502044048634678543233616265407 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.961295342052216983850119223211706034603 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1}{z} + 1\right) \cdot \frac{t \cdot x}{z} + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.387575495283554302798986221595555673559 \cdot 10^{280}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))))