Average Error: 4.6 → 0.6
Time: 5.6s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.1268122843286182 \cdot 10^{-194}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{y - z \cdot \left(y + t\right)}{1 - z}}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.0165756137896936 \cdot 10^{+308}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z \cdot \left(y + t\right)\right) \cdot \frac{x}{z}}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\
\;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.1268122843286182 \cdot 10^{-194}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{y - z \cdot \left(y + t\right)}{1 - z}}}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.0165756137896936 \cdot 10^{+308}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

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

\end{array}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (- (/ y z) (/ t (- 1.0 z))) (- INFINITY))
   (/ (* x (- y (* z (+ y t)))) (* z (- 1.0 z)))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) -1.1268122843286182e-194)
     (* (- (/ y z) (/ t (- 1.0 z))) x)
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 0.0)
       (/ (/ x z) (/ 1.0 (/ (- y (* z (+ y t))) (- 1.0 z))))
       (if (<= (- (/ y z) (/ t (- 1.0 z))) 1.0165756137896936e+308)
         (* (- (/ y z) (/ t (- 1.0 z))) x)
         (/ (* (- y (* z (+ y t))) (/ x z)) (- 1.0 z)))))))
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 tmp;
	if (((y / z) - (t / (1.0 - z))) <= -((double) INFINITY)) {
		tmp = (x * (y - (z * (y + t)))) / (z * (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= -1.1268122843286182e-194) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else if (((y / z) - (t / (1.0 - z))) <= 0.0) {
		tmp = (x / z) / (1.0 / ((y - (z * (y + t))) / (1.0 - z)));
	} else if (((y / z) - (t / (1.0 - z))) <= 1.0165756137896936e+308) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else {
		tmp = ((y - (z * (y + t))) * (x / z)) / (1.0 - z);
	}
	return tmp;
}

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
Target4.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 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-sub_binary6464.0

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

    4. Applied associate-*r/_binary640.3

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

    5. Simplified0.3

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

    if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -1.12681228432861822e-194 or 0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.01657561378969364e308

    1. Initial program 0.2

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

    if -1.12681228432861822e-194 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 0.0

    1. Initial program 11.0

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

    3. Applied frac-sub_binary6423.9

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

    4. Applied associate-*r/_binary6422.4

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

    5. Simplified22.4

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

    7. Applied associate-/l*_binary6424.0

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

    8. Simplified15.4

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

    10. Applied div-inv_binary6415.4

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

    11. Applied associate-/r*_binary644.1

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

    if 1.01657561378969364e308 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 64.0

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

    3. Applied frac-sub_binary6464.0

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

    4. Applied associate-*r/_binary640.3

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

    5. Simplified0.3

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

    7. Applied associate-/r*_binary640.3

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

    8. Simplified0.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot \left(y + t\right)\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -1.1268122843286182 \cdot 10^{-194}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{y - z \cdot \left(y + t\right)}{1 - z}}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.0165756137896936 \cdot 10^{+308}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z \cdot \left(y + t\right)\right) \cdot \frac{x}{z}}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(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 (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))