Average Error: 4.4 → 2.2
Time: 14.4s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -1.7626926400260635 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 0 \lor \neg \left(x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 1.5691824587696755 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x \cdot \frac{t}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -1.7626926400260635 \cdot 10^{-229}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\

\mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 0 \lor \neg \left(x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 1.5691824587696755 \cdot 10^{+303}\right):\\
\;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y}{z} - x \cdot \frac{t}{1 - z}\\

\end{array}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (* x (- (/ y z) (/ t (- 1.0 z)))) -1.7626926400260635e-229)
   (* x (- (/ y z) (/ t (- 1.0 z))))
   (if (or (<= (* x (- (/ y z) (/ t (- 1.0 z)))) 0.0)
           (not
            (<= (* x (- (/ y z) (/ t (- 1.0 z)))) 1.5691824587696755e+303)))
     (- (/ (* x y) z) (/ (* x t) (- 1.0 z)))
     (- (* x (/ y z)) (* x (/ t (- 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 ((x * ((y / z) - (t / (1.0 - z)))) <= -1.7626926400260635e-229) {
		tmp = x * ((y / z) - (t / (1.0 - z)));
	} else if (((x * ((y / z) - (t / (1.0 - z)))) <= 0.0) || !((x * ((y / z) - (t / (1.0 - z)))) <= 1.5691824587696755e+303)) {
		tmp = ((x * y) / z) - ((x * t) / (1.0 - z));
	} else {
		tmp = (x * (y / z)) - (x * (t / (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
Target4.2
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < -1.76269264002606355e-229

    1. Initial program 4.2

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

    if -1.76269264002606355e-229 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 0.0 or 1.5691824587696755e303 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))))

    1. Initial program 14.2

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

    3. Applied add-cube-cbrt_binary64_1102514.4

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

    4. Taylor expanded around 0 2.0

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

    if 0.0 < (*.f64 x (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))) < 1.5691824587696755e303

    1. Initial program 0.3

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

    3. Applied sub-neg_binary64_109830.3

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

    4. Applied distribute-rgt-in_binary64_109400.3

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

    5. Simplified0.3

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

    6. Simplified0.3

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq -1.7626926400260635 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 0 \lor \neg \left(x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \leq 1.5691824587696755 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x \cdot \frac{t}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))