Average Error: 4.2 → 0.7
Time: 4.2s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.9683893696444749 \cdot 10^{121}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.59090119726614977 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.38430042432179337 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.44659480199283158 \cdot 10^{218}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.9683893696444749 \cdot 10^{121}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.59090119726614977 \cdot 10^{-199}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.38430042432179337 \cdot 10^{-257}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.44659480199283158 \cdot 10^{218}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + \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 ((((y / z) - (t / (1.0 - z))) <= -1.9683893696444749e+121)) {
		VAR = ((((x * y) / z) + -(x * (t / (1.0 - z)))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
	} else {
		double VAR_1;
		if ((((y / z) - (t / (1.0 - z))) <= -8.59090119726615e-199)) {
			VAR_1 = (x * ((y / z) - (t * (1.0 / (1.0 - z)))));
		} else {
			double VAR_2;
			if ((((y / z) - (t / (1.0 - z))) <= 1.3843004243217934e-257)) {
				VAR_2 = (fma(y, (x / z), fma(1.0, ((t * x) / pow(z, 2.0)), ((t * x) / z))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
			} else {
				double VAR_3;
				if ((((y / z) - (t / (1.0 - z))) <= 2.4465948019928316e+218)) {
					VAR_3 = (x * ((y / z) - (t * (1.0 / (1.0 - z)))));
				} else {
					VAR_3 = ((((x * y) / z) + -(x * (t / (1.0 - z)))) + (x * ((t / (1.0 - z)) * (-1.0 + 1.0))));
				}
				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

Original4.2
Target4.1
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 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -1.9683893696444749e+121 or 2.4465948019928316e+218 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 14.0

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

    3. Applied add-cube-cbrt14.4

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

    4. Applied div-inv14.5

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

    5. Applied prod-diff14.5

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

    6. Applied distribute-lft-in14.5

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

    7. Simplified14.1

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

    8. Simplified14.1

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

    10. Applied fma-udef14.1

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

    11. Applied distribute-lft-in14.1

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

    12. Simplified1.7

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

    13. Simplified1.7

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

    if -1.9683893696444749e+121 < (- (/ y z) (/ t (- 1.0 z))) < -8.59090119726615e-199 or 1.3843004243217934e-257 < (- (/ y z) (/ t (- 1.0 z))) < 2.4465948019928316e+218

    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

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

    if -8.59090119726615e-199 < (- (/ y z) (/ t (- 1.0 z))) < 1.3843004243217934e-257

    1. Initial program 9.0

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

    3. Applied add-cube-cbrt9.2

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

    4. Applied div-inv9.2

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

    5. Applied prod-diff9.2

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

    6. Applied distribute-lft-in9.2

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

    7. Simplified9.1

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

    8. Simplified9.1

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

    9. Taylor expanded around inf 1.2

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

    10. Simplified1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.9683893696444749 \cdot 10^{121}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.59090119726614977 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.38430042432179337 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.44659480199283158 \cdot 10^{218}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x \cdot \frac{t}{1 - z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))))