Average Error: 4.7 → 1.4
Time: 16.8s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.704089819968152 \cdot 10^{+297}:\\ \;\;\;\;\frac{y \cdot x}{z} - \left(t \cdot \left(z \cdot x\right) + t \cdot x\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.4057585209161635 \cdot 10^{+189}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.704089819968152 \cdot 10^{+297}:\\
\;\;\;\;\frac{y \cdot x}{z} - \left(t \cdot \left(z \cdot x\right) + t \cdot x\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.4057585209161635 \cdot 10^{+189}:\\
\;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{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 (<= (- (/ y z) (/ t (- 1.0 z))) -7.704089819968152e+297)
   (- (/ (* y x) z) (+ (* t (* z x)) (* t x)))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) 1.4057585209161635e+189)
     (- (* (/ y z) x) (* (/ t (- 1.0 z)) x))
     (- (/ (* y x) z) (/ (* t x) (- 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))) <= -7.704089819968152e+297) {
		tmp = ((y * x) / z) - ((t * (z * x)) + (t * x));
	} else if (((y / z) - (t / (1.0 - z))) <= 1.4057585209161635e+189) {
		tmp = ((y / z) * x) - ((t / (1.0 - z)) * x);
	} else {
		tmp = ((y * x) / z) - ((t * x) / (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

Original4.7
Target4.3
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -7.70408981996815247e297

    1. Initial program 51.7

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

    2. Taylor expanded around 0 0.3

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

    if -7.70408981996815247e297 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.40575852091616354e189

    1. Initial program 1.5

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

    3. Applied sub-neg_binary64_120061.5

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

    4. Applied distribute-rgt-in_binary64_119631.5

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

    if 1.40575852091616354e189 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 18.5

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

    2. Taylor expanded around 0 1.0

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -7.704089819968152 \cdot 10^{+297}:\\ \;\;\;\;\frac{y \cdot x}{z} - \left(t \cdot \left(z \cdot x\right) + t \cdot x\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.4057585209161635 \cdot 10^{+189}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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)))))