Average Error: 4.8 → 3.1
Time: 9.3s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -7.5792827577132941 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.594648028 \cdot 10^{-315}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]
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 \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -7.5792827577132941 \cdot 10^{-140}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.594648028 \cdot 10^{-315}:\\
\;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\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 temp;
	if ((((y / z) - (t / (1.0 - z))) <= -inf.0)) {
		temp = ((x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z)));
	} else {
		double temp_1;
		if ((((y / z) - (t / (1.0 - z))) <= -7.579282757713294e-140)) {
			temp_1 = (x * ((y / z) - (1.0 / ((1.0 - z) / t))));
		} else {
			double temp_2;
			if ((((y / z) - (t / (1.0 - z))) <= 1.5946480275096e-315)) {
				temp_2 = fma(x, (y / z), fma(1.0, ((t * x) / pow(z, 2.0)), ((t * x) / z)));
			} else {
				temp_2 = (x * ((y / z) - (1.0 / ((1.0 - z) / t))));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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
Target4.4
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

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

    3. Applied frac-sub64.0

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

    4. Applied associate-*r/0.2

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -7.579282757713294e-140 or 1.5946480275096e-315 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 2.4

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

    3. Applied clear-num2.4

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

    if -7.579282757713294e-140 < (- (/ y z) (/ t (- 1.0 z))) < 1.5946480275096e-315

    1. Initial program 8.5

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified7.9

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -7.5792827577132941 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.594648028 \cdot 10^{-315}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))))