Average Error: 4.8 → 0.9
Time: 16.6s
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 r258183 = x;
        double r258184 = y;
        double r258185 = z;
        double r258186 = r258184 / r258185;
        double r258187 = t;
        double r258188 = 1.0;
        double r258189 = r258188 - r258185;
        double r258190 = r258187 / r258189;
        double r258191 = r258186 - r258190;
        double r258192 = r258183 * r258191;
        return r258192;
}
↓
double f(double x, double y, double z, double t) {
        double r258193 = y;
        double r258194 = z;
        double r258195 = r258193 / r258194;
        double r258196 = t;
        double r258197 = 1.0;
        double r258198 = r258197 - r258194;
        double r258199 = r258196 / r258198;
        double r258200 = r258195 - r258199;
        double r258201 = -1.3561375991187766e+280;
        bool r258202 = r258200 <= r258201;
        double r258203 = x;
        double r258204 = r258193 * r258198;
        double r258205 = r258194 * r258196;
        double r258206 = r258204 - r258205;
        double r258207 = r258203 * r258206;
        double r258208 = r258194 * r258198;
        double r258209 = r258207 / r258208;
        double r258210 = -5.000777057502044e-118;
        bool r258211 = r258200 <= r258210;
        double r258212 = 1.0;
        double r258213 = r258198 / r258196;
        double r258214 = r258212 / r258213;
        double r258215 = r258195 - r258214;
        double r258216 = r258203 * r258215;
        double r258217 = 5.961295342052217e-199;
        bool r258218 = r258200 <= r258217;
        double r258219 = r258197 / r258194;
        double r258220 = r258219 + r258212;
        double r258221 = r258196 * r258203;
        double r258222 = r258221 / r258194;
        double r258223 = r258220 * r258222;
        double r258224 = r258203 * r258193;
        double r258225 = r258224 / r258194;
        double r258226 = r258223 + r258225;
        double r258227 = 1.3875754952835543e+280;
        bool r258228 = r258200 <= r258227;
        double r258229 = r258228 ? r258216 : r258209;
        double r258230 = r258218 ? r258226 : r258229;
        double r258231 = r258211 ? r258216 : r258230;
        double r258232 = r258202 ? r258209 : r258231;
        return r258232;
}

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)))))