Average Error: 4.4 → 0.2
Time: 5.0s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.1983901674519873 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.539867945929258 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.4407411231252675 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + \left(y + \frac{t}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.5237804266397524 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.1983901674519873 \cdot 10^{+307}:\\
\;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.539867945929258 \cdot 10^{-308}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.4407411231252675 \cdot 10^{-289}:\\
\;\;\;\;\frac{x}{z} \cdot \left(t + \left(y + \frac{t}{z}\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.5237804266397524 \cdot 10^{+306}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

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

\end{array}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (- (/ y z) (/ t (- 1.0 z))) -4.1983901674519873e+307)
   (/ (* x (- y (* z (+ y t)))) (* z (- 1.0 z)))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) -4.539867945929258e-308)
     (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 2.4407411231252675e-289)
       (* (/ x z) (+ t (+ y (/ t z))))
       (if (<= (- (/ y z) (/ t (- 1.0 z))) 2.5237804266397524e+306)
         (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))
         (/ (* x (- y (* z (+ y t)))) (* z (- 1.0 z))))))))
double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((y / z) - (t / (1.0 - z))) <= -4.1983901674519873e+307) {
		tmp = (x * (y - (z * (y + t)))) / (z * (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= -4.539867945929258e-308) {
		tmp = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	} else if (((y / z) - (t / (1.0 - z))) <= 2.4407411231252675e-289) {
		tmp = (x / z) * (t + (y + (t / z)));
	} else if (((y / z) - (t / (1.0 - z))) <= 2.5237804266397524e+306) {
		tmp = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	} else {
		tmp = (x * (y - (z * (y + t)))) / (z * (1.0 - z));
	}
	return tmp;
}

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

Original4.4
Target4.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.1983901674519873e307 or 2.52378042663975236e306 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 62.2

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

    3. Applied frac-sub_binary6462.2

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

    4. Applied associate-*r/_binary640.3

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

    5. Simplified0.3

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

    if -4.1983901674519873e307 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.5398679459292579e-308 or 2.4407411231252675e-289 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.52378042663975236e306

    1. Initial program 0.2

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

    3. Applied div-inv_binary640.2

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

    if -4.5398679459292579e-308 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.4407411231252675e-289

    1. Initial program 16.1

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.1983901674519873 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.539867945929258 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.4407411231252675 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{z} \cdot \left(t + \left(y + \frac{t}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2.5237804266397524 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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 (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))