Average Error: 0.1 → 0.1
Time: 9.0s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(z - \left(y \cdot \log \left(\frac{1}{y}\right) + \left(y + \log \left(\frac{1}{y}\right) \cdot 0.5\right)\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(z - \left(y \cdot \log \left(\frac{1}{y}\right) + \left(y + \log \left(\frac{1}{y}\right) \cdot 0.5\right)\right)\right)
(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z)
 :precision binary64
 (- x (- z (+ (* y (log (/ 1.0 y))) (+ y (* (log (/ 1.0 y)) 0.5))))))
double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}
↓
double code(double x, double y, double z) {
	return x - (z - ((y * log(1.0 / y)) + (y + (log(1.0 / y) * 0.5))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
  2. Using strategy rm

  3. Applied associate-+l-_binary64_116070.1

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

  4. Applied associate--l-_binary64_116100.1

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

  5. Simplified0.1

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

  6. Taylor expanded around inf 0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2021096 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))