Average Error: 4.4 → 0.4
Time: 7.3s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.62330331140472958 \cdot 10^{-267}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.40791248403575632 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.2292296759795089 \cdot 10^{175}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\
\;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.62330331140472958 \cdot 10^{-267}:\\
\;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.40791248403575632 \cdot 10^{-220}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.2292296759795089 \cdot 10^{175}:\\
\;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z))))))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -inf.0)) {
		VAR = ((double) (((double) (((double) (x * y)) / z)) + ((double) (x * ((double) -(((double) (t / ((double) (1.0 - z))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -2.6233033114047296e-267)) {
			VAR_1 = ((double) (x * ((double) fma(y, ((double) (1.0 / z)), ((double) -(((double) (t / ((double) (1.0 - z))))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 7.407912484035756e-220)) {
				VAR_2 = ((double) fma(y, ((double) (x / z)), ((double) fma(1.0, ((double) (((double) (t * x)) / ((double) pow(z, 2.0)))), ((double) (((double) (t * x)) / z))))));
			} else {
				double VAR_3;
				if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 7.229229675979509e+175)) {
					VAR_3 = ((double) (x * ((double) fma(y, ((double) (1.0 / z)), ((double) -(((double) (t / ((double) (1.0 - z))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) (x * y)) / z)) + ((double) (x * ((double) -(((double) (t / ((double) (1.0 - z))))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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

Original4.4
Target4.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 7.229229675979509e+175 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 23.7

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

    3. Applied div-inv23.8

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

    4. Applied fma-neg23.8

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

    6. Applied fma-udef23.8

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

    7. Applied distribute-lft-in23.8

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

    8. Simplified0.9

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -2.6233033114047296e-267 or 7.407912484035756e-220 < (- (/ y z) (/ t (- 1.0 z))) < 7.229229675979509e+175

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    4. Applied fma-neg0.3

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

    if -2.6233033114047296e-267 < (- (/ y z) (/ t (- 1.0 z))) < 7.407912484035756e-220

    1. Initial program 10.9

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

    3. Applied div-inv10.9

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

    4. Applied fma-neg10.9

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

    5. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

    6. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.62330331140472958 \cdot 10^{-267}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.40791248403575632 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.2292296759795089 \cdot 10^{175}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +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)))))