Average Error: 25.7 → 9.6
Time: 8.8s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.8866702549812757 \cdot 10^{-93}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)}}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.8866702549812757 \cdot 10^{-93}:\\
\;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\

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

\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 -2.8866702549812757e-93)
   (- x (/ (log (- 1.0 (- y (* y (exp z))))) t))
   (- x (/ y (/ t (+ z (* (- 1.0 y) (* (* z z) 0.5))))))))
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 <= -2.8866702549812757e-93) {
		tmp = x - (log(1.0 - (y - (y * exp(z)))) / t);
	} else {
		tmp = x - (y / (t / (z + ((1.0 - y) * ((z * z) * 0.5)))));
	}
	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

Original25.7
Target17.0
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.88667025498127572e-93

    1. Initial program 16.1

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

    3. Applied associate-+l-_binary64_887913.9

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

    if -2.88667025498127572e-93 < z

    1. Initial program 32.0

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified7.7

      \[\leadsto x - \frac{\color{blue}{y \cdot z + \left(0.5 \cdot \left(y \cdot \left(z \cdot z\right)\right)\right) \cdot \left(1 - y\right)}}{t}\]
    4. Using strategy rm

    5. Applied clear-num_binary64_89437.7

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

    6. Simplified7.7

      \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{y \cdot \left(z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)\right)}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity_binary64_89447.7

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

    9. Applied add-sqr-sqrt_binary64_89667.7

      \[\leadsto x - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \frac{t}{y \cdot \left(z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)\right)}}\]

    10. Applied times-frac_binary64_89507.7

      \[\leadsto x - \color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\frac{t}{y \cdot \left(z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)\right)}}}\]

    11. Simplified7.7

      \[\leadsto x - \color{blue}{1} \cdot \frac{\sqrt{1}}{\frac{t}{y \cdot \left(z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)\right)}}\]

    12. Simplified6.7

      \[\leadsto x - 1 \cdot \color{blue}{\frac{y}{\frac{t}{z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8866702549812757 \cdot 10^{-93}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z + \left(1 - y\right) \cdot \left(\left(z \cdot z\right) \cdot 0.5\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(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)))