Average Error: 4.8 → 2.4
Time: 4.8s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.7780768828379175 \cdot 10^{+214} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 6.576784828256989 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{1 - z}}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.7780768828379175 \cdot 10^{+214} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 6.576784828256989 \cdot 10^{+223}\right):\\
\;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{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 (or (<= (- (/ y z) (/ t (- 1.0 z))) -1.7780768828379175e+214)
         (not (<= (- (/ y z) (/ t (- 1.0 z))) 6.576784828256989e+223)))
   (/ (* x (- y (* z (+ y t)))) (* z (- 1.0 z)))
   (*
    x
    (-
     (/ y z)
     (*
      (/ (* (cbrt t) (cbrt t)) (* (cbrt (- 1.0 z)) (cbrt (- 1.0 z))))
      (/ (cbrt t) (cbrt (- 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))) <= -1.7780768828379175e+214) || !(((y / z) - (t / (1.0 - z))) <= 6.576784828256989e+223)) {
		tmp = (x * (y - (z * (y + t)))) / (z * (1.0 - z));
	} else {
		tmp = x * ((y / z) - (((cbrt(t) * cbrt(t)) / (cbrt(1.0 - z) * cbrt(1.0 - z))) * (cbrt(t) / cbrt(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

Original4.8
Target4.3
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.7780768828379175e214 or 6.57678482825698866e223 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 22.9

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

    3. Applied frac-sub_binary6425.9

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

    4. Applied associate-*r/_binary643.8

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

    5. Simplified3.8

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

    if -1.7780768828379175e214 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 6.57678482825698866e223

    1. Initial program 1.7

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

    3. Applied add-cube-cbrt_binary641.9

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

    4. Applied add-cube-cbrt_binary642.2

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

    5. Applied times-frac_binary642.2

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.7780768828379175 \cdot 10^{+214} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 6.576784828256989 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{1 - z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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)))))