Average Error: 4.8 → 3.7
Time: 2.9s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.6308762846363051 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;z \le 5.9956792597095237 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(-y\right) \cdot \left(1 - z\right), x \cdot \left(t \cdot z\right)\right)}{\left(-z\right) \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-t}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.6308762846363051 \cdot 10^{-16}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

\mathbf{elif}\;z \le 5.9956792597095237 \cdot 10^{-116}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, \left(-y\right) \cdot \left(1 - z\right), x \cdot \left(t \cdot z\right)\right)}{\left(-z\right) \cdot \left(1 - z\right)}\\

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

\end{array}
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 VAR;
	if ((z <= -2.630876284636305e-16)) {
		VAR = (x * ((y / z) - (t * (1.0 / (1.0 - z)))));
	} else {
		double VAR_1;
		if ((z <= 5.995679259709524e-116)) {
			VAR_1 = (fma(x, (-y * (1.0 - z)), (x * (t * z))) / (-z * (1.0 - z)));
		} else {
			VAR_1 = (x * ((y / z) + (-t / (1.0 - z))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Herbie3.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.630876284636305e-16

    1. Initial program 2.2

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

    3. Applied div-inv2.3

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

    if -2.630876284636305e-16 < z < 5.995679259709524e-116

    1. Initial program 10.0

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

    3. Applied div-inv10.0

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

    5. Applied un-div-inv10.0

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

    6. Applied frac-2neg10.0

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

    7. Applied frac-sub10.0

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

    8. Applied associate-*r/6.3

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

    9. Simplified6.3

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

    if 5.995679259709524e-116 < z

    1. Initial program 2.6

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

    3. Applied sub-neg2.6

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

    4. Simplified2.6

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.6308762846363051 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;z \le 5.9956792597095237 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(-y\right) \cdot \left(1 - z\right), x \cdot \left(t \cdot z\right)\right)}{\left(-z\right) \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{-t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))