Average Error: 4.7 → 1.1
Time: 4.9s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.579993657727699 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}} - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5.53040647017809 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.2385761353250375 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 8.063107108367971 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.579993657727699 \cdot 10^{+274}:\\
\;\;\;\;\frac{1}{\frac{z}{y \cdot x}} - \frac{t}{1 - z} \cdot x\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5.53040647017809 \cdot 10^{-168}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.2385761353250375 \cdot 10^{-236}:\\
\;\;\;\;\frac{y}{z} \cdot x - \frac{t \cdot x}{1 - z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 8.063107108367971 \cdot 10^{+142}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\

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

\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))) -7.579993657727699e+274)
   (- (/ 1.0 (/ z (* y x))) (* (/ t (- 1.0 z)) x))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) -5.53040647017809e-168)
     (* x (+ (/ y z) (/ -1.0 (/ (- 1.0 z) t))))
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 4.2385761353250375e-236)
       (- (* (/ y z) x) (/ (* t x) (- 1.0 z)))
       (if (<= (- (/ y z) (/ t (- 1.0 z))) 8.063107108367971e+142)
         (* x (+ (/ y z) (/ -1.0 (/ (- 1.0 z) t))))
         (- (* y (/ x z)) (* (/ t (- 1.0 z)) x)))))))
double code(double x, double y, double z, double t) {
	return ((double) (x * ((double) ((y / z) - (t / ((double) (1.0 - z)))))));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((((double) ((y / z) - (t / ((double) (1.0 - z))))) <= -7.579993657727699e+274)) {
		tmp = ((double) ((1.0 / (z / ((double) (y * x)))) - ((double) ((t / ((double) (1.0 - z))) * x))));
	} else {
		double tmp_1;
		if ((((double) ((y / z) - (t / ((double) (1.0 - z))))) <= -5.53040647017809e-168)) {
			tmp_1 = ((double) (x * ((double) ((y / z) + (-1.0 / (((double) (1.0 - z)) / t))))));
		} else {
			double tmp_2;
			if ((((double) ((y / z) - (t / ((double) (1.0 - z))))) <= 4.2385761353250375e-236)) {
				tmp_2 = ((double) (((double) ((y / z) * x)) - (((double) (t * x)) / ((double) (1.0 - z)))));
			} else {
				double tmp_3;
				if ((((double) ((y / z) - (t / ((double) (1.0 - z))))) <= 8.063107108367971e+142)) {
					tmp_3 = ((double) (x * ((double) ((y / z) + (-1.0 / (((double) (1.0 - z)) / t))))));
				} else {
					tmp_3 = ((double) (((double) (y * (x / z))) - ((double) ((t / ((double) (1.0 - z))) * x))));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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

Original4.7
Target4.3
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -7.5799936577276988e274

    1. Initial program Error: 38.3 bits

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

    3. Applied sub-negError: 38.3 bits

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

    4. Applied distribute-lft-inError: 38.3 bits

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

    5. SimplifiedError: 38.3 bits

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \frac{-t}{1 - z}}\]
    6. Using strategy rm

    7. Applied associate-*r/Error: 0.4 bits

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \frac{-t}{1 - z}\]
    8. Using strategy rm

    9. Applied clear-numError: 0.4 bits

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

    if -7.5799936577276988e274 < (- (/ y z) (/ t (- 1.0 z))) < -5.53040647017808971e-168 or 4.23857613532503754e-236 < (- (/ y z) (/ t (- 1.0 z))) < 8.06310710836797102e142

    1. Initial program Error: 0.2 bits

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

    3. Applied clear-numError: 0.3 bits

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

    if -5.53040647017808971e-168 < (- (/ y z) (/ t (- 1.0 z))) < 4.23857613532503754e-236

    1. Initial program Error: 8.2 bits

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

    3. Applied sub-negError: 8.2 bits

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

    4. Applied distribute-lft-inError: 8.2 bits

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

    5. SimplifiedError: 8.2 bits

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \frac{-t}{1 - z}}\]
    6. Using strategy rm

    7. Applied associate-*r/Error: 4.4 bits

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

    if 8.06310710836797102e142 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program Error: 13.2 bits

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

    3. Applied sub-negError: 13.2 bits

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

    4. Applied distribute-lft-inError: 13.2 bits

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

    5. SimplifiedError: 13.2 bits

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \frac{-t}{1 - z}}\]
    6. Using strategy rm

    7. Applied associate-*r/Error: 1.4 bits

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

    8. Taylor expanded around 0 Error: 1.4 bits

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

    9. SimplifiedError: 1.8 bits

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} + x \cdot \frac{-t}{1 - z}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 1.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.579993657727699 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}} - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -5.53040647017809 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.2385761353250375 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 8.063107108367971 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))))