Average Error: 4.6 → 0.7
Time: 12.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \frac{1 - z}{\sqrt[3]{t}} - z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}{z \cdot \frac{1 - z}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.5300376642230815 \cdot 10^{-281}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40252814846900272 \cdot 10^{-189}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.0414771802092237 \cdot 10^{300}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\left(y \cdot \left(1 - z\right) - z \cdot t\right)\right)}{\left(-z\right) \cdot \left(1 - z\right)}\\ \end{array}\]
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 \left(y \cdot \frac{1 - z}{\sqrt[3]{t}} - z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}{z \cdot \frac{1 - z}{\sqrt[3]{t}}}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.5300376642230815 \cdot 10^{-281}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40252814846900272 \cdot 10^{-189}:\\
\;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.0414771802092237 \cdot 10^{300}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(-\left(y \cdot \left(1 - z\right) - z \cdot t\right)\right)}{\left(-z\right) \cdot \left(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) (x * ((double) (((double) (y * ((double) (((double) (1.0 - z)) / ((double) cbrt(t)))))) - ((double) (z * ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))))))) / ((double) (z * ((double) (((double) (1.0 - z)) / ((double) cbrt(t))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -1.5300376642230815e-281)) {
			VAR_1 = ((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 2.4025281484690027e-189)) {
				VAR_2 = ((double) (((double) (((double) (x * y)) / z)) + ((double) (((double) (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)))))) <= 8.041477180209224e+300)) {
					VAR_3 = ((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z))))))));
				} else {
					VAR_3 = ((double) (((double) (x * ((double) -(((double) (((double) (y * ((double) (1.0 - z)))) - ((double) (z * t)))))))) / ((double) (((double) -(z)) * ((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

Original4.6
Target4.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0

    1. Initial program 64.0

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

    3. Applied add-cube-cbrt64.0

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

    4. Applied associate-/l*64.0

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

    5. Applied frac-sub64.0

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

    6. Applied associate-*r/15.2

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -1.5300376642230815e-281 or 2.4025281484690027e-189 < (- (/ y z) (/ t (- 1.0 z))) < 8.041477180209224e+300

    1. Initial program 0.2

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

    if -1.5300376642230815e-281 < (- (/ y z) (/ t (- 1.0 z))) < 2.4025281484690027e-189

    1. Initial program 10.0

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

    2. Taylor expanded around inf 1.2

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

    if 8.041477180209224e+300 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 56.3

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

    3. Applied frac-2neg56.3

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

    4. Applied frac-sub56.3

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

    5. Applied associate-*r/0.3

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

    6. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \frac{1 - z}{\sqrt[3]{t}} - z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}{z \cdot \frac{1 - z}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.5300376642230815 \cdot 10^{-281}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40252814846900272 \cdot 10^{-189}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.0414771802092237 \cdot 10^{300}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\left(y \cdot \left(1 - z\right) - z \cdot t\right)\right)}{\left(-z\right) \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(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)))))