Average Error: 4.8 → 4.6
Time: 6.2s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0011343725349988 \cdot 10^{191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, -x \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \le -0.8424720202832825:\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + \left(-\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot x\right) \cdot \frac{\sqrt[3]{t}}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \le 8.1853912102024854 \cdot 10^{-67}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{t}{1 - z}\right)\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{z}}{\frac{1}{y}} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0011343725349988 \cdot 10^{191}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, -x \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\

\mathbf{elif}\;x \le -0.8424720202832825:\\
\;\;\;\;\left(\frac{x}{\frac{z}{y}} + \left(-\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot x\right) \cdot \frac{\sqrt[3]{t}}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\

\mathbf{elif}\;x \le 8.1853912102024854 \cdot 10^{-67}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + \left(-\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{t}{1 - z}\right)\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{x}{z}}{\frac{1}{y}} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\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 ((x <= -1.0011343725349988e+191)) {
		VAR = (fma((x / z), y, -(x * (t / (1.0 - z)))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
	} else {
		double VAR_1;
		if ((x <= -0.8424720202832825)) {
			VAR_1 = (((x / (z / y)) + -(((cbrt(t) * cbrt(t)) * x) * (cbrt(t) / (1.0 - z)))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
		} else {
			double VAR_2;
			if ((x <= 8.185391210202485e-67)) {
				VAR_2 = ((((x * y) / z) + -((cbrt(x) * cbrt(x)) * (cbrt(x) * (t / (1.0 - z))))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
			} else {
				VAR_2 = ((((x / z) / (1.0 / y)) + -(x * (t / (1.0 - z)))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
			}
			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

Original4.8
Target4.3
Herbie4.6
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.0011343725349988e+191

    1. Initial program 6.5

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

    3. Applied add-cube-cbrt7.0

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

    4. Applied div-inv7.0

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

    5. Applied prod-diff7.0

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

    6. Applied distribute-lft-in7.0

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

    7. Simplified6.6

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

    8. Simplified6.5

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

    10. Applied fma-udef6.5

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

    11. Applied distribute-lft-in6.5

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

    12. Simplified16.8

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

    13. Simplified16.8

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

    15. Applied associate-/l*6.5

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

    17. Applied associate-/r/13.7

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

    18. Applied fma-def13.7

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

    if -1.0011343725349988e+191 < x < -0.8424720202832825

    1. Initial program 2.0

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

    3. Applied add-cube-cbrt2.4

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

    4. Applied div-inv2.5

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

    5. Applied prod-diff2.5

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

    6. Applied distribute-lft-in2.5

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

    7. Simplified2.0

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

    8. Simplified2.0

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

    10. Applied fma-udef2.0

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

    11. Applied distribute-lft-in2.0

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

    12. Simplified7.3

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

    13. Simplified7.3

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

    15. Applied associate-/l*2.1

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

    17. Applied *-un-lft-identity2.1

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

    18. Applied add-cube-cbrt2.6

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

    19. Applied times-frac2.6

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

    20. Applied associate-*r*2.9

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

    21. Simplified2.9

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

    if -0.8424720202832825 < x < 8.185391210202485e-67

    1. Initial program 6.1

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

    3. Applied add-cube-cbrt6.5

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

    4. Applied div-inv6.6

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

    5. Applied prod-diff6.6

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

    6. Applied distribute-lft-in6.6

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

    7. Simplified6.2

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

    8. Simplified6.2

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

    10. Applied fma-udef6.2

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

    11. Applied distribute-lft-in6.2

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

    12. Simplified2.9

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

    13. Simplified2.9

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

    15. Applied add-cube-cbrt3.3

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

    16. Applied associate-*l*3.3

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

    if 8.185391210202485e-67 < x

    1. Initial program 3.3

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

    3. Applied add-cube-cbrt3.8

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

    4. Applied div-inv3.8

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

    5. Applied prod-diff3.8

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

    6. Applied distribute-lft-in3.8

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

    7. Simplified3.3

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

    8. Simplified3.3

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

    10. Applied fma-udef3.3

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

    11. Applied distribute-lft-in3.3

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

    12. Simplified7.3

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

    13. Simplified7.3

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

    15. Applied associate-/l*3.4

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

    17. Applied div-inv3.4

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

    18. Applied associate-/r*6.5

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0011343725349988 \cdot 10^{191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, -x \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \le -0.8424720202832825:\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + \left(-\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot x\right) \cdot \frac{\sqrt[3]{t}}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \le 8.1853912102024854 \cdot 10^{-67}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{t}{1 - z}\right)\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{z}}{\frac{1}{y}} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]

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)))))