Average Error: 25.8 → 12.2
Time: 9.3s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -4.583227701693125 \cdot 10^{-26}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \leq -1.566036451166551 \cdot 10^{-278}:\\ \;\;\;\;x - \frac{z \cdot y + \left(0.5 \cdot \left(z \cdot z\right)\right) \cdot \left(y - y \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(z \cdot z\right) \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -4.583227701693125 \cdot 10^{-26}:\\
\;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\

\mathbf{elif}\;z \leq -1.566036451166551 \cdot 10^{-278}:\\
\;\;\;\;x - \frac{z \cdot y + \left(0.5 \cdot \left(z \cdot z\right)\right) \cdot \left(y - y \cdot y\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(z \cdot z\right) \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)}{t}\\

\end{array}
(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.583227701693125e-26)
   (- x (/ (log (- 1.0 (- y (* y (exp z))))) t))
   (if (<= z -1.566036451166551e-278)
     (- x (/ (+ (* z y) (* (* 0.5 (* z z)) (- y (* y y)))) t))
     (-
      x
      (/
       (log (+ 1.0 (* y (+ z (* (* z z) (+ 0.5 (* z 0.16666666666666666)))))))
       t)))))
double code(double x, double y, double z, double t) {
	return x - (log((1.0 - y) + (y * exp(z))) / t);
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.583227701693125e-26) {
		tmp = x - (log(1.0 - (y - (y * exp(z)))) / t);
	} else if (z <= -1.566036451166551e-278) {
		tmp = x - (((z * y) + ((0.5 * (z * z)) * (y - (y * y)))) / t);
	} else {
		tmp = x - (log(1.0 + (y * (z + ((z * z) * (0.5 + (z * 0.16666666666666666)))))) / t);
	}
	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

Original25.8
Target16.3
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.58322770169312487e-26

    1. Initial program 12.0

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

    3. Applied associate-+l-_binary64_1046111.7

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

    if -4.58322770169312487e-26 < z < -1.56603645116655093e-278

    1. Initial program 32.1

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

    2. Taylor expanded around 0 13.3

      \[\leadsto x - \frac{\color{blue}{\left(0.5 \cdot \left({z}^{2} \cdot y\right) + z \cdot y\right) - 0.5 \cdot \left({z}^{2} \cdot {y}^{2}\right)}}{t}\]

    3. Simplified13.3

      \[\leadsto x - \frac{\color{blue}{y \cdot z + \left(0.5 \cdot \left(z \cdot z\right)\right) \cdot \left(y - y \cdot y\right)}}{t}\]

    if -1.56603645116655093e-278 < z

    1. Initial program 32.3

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

    3. Applied associate-+l-_binary64_1046115.9

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

    4. Taylor expanded around 0 11.8

      \[\leadsto x - \frac{\log \left(1 - \color{blue}{\left(-\left(0.5 \cdot \left({z}^{2} \cdot y\right) + \left(0.16666666666666666 \cdot \left({z}^{3} \cdot y\right) + z \cdot y\right)\right)\right)}\right)}{t}\]

    5. Simplified11.8

      \[\leadsto x - \frac{\log \left(1 - \color{blue}{\left(-y \cdot \left(z + \left(z \cdot z\right) \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)}\right)}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.583227701693125 \cdot 10^{-26}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \leq -1.566036451166551 \cdot 10^{-278}:\\ \;\;\;\;x - \frac{z \cdot y + \left(0.5 \cdot \left(z \cdot z\right)\right) \cdot \left(y - y \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(z \cdot z\right) \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))