Average Error: 2.8 → 0.0
Time: 10.4s
Precision: binary64
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{-1}{x + \frac{e^{z} \cdot -1.1283791670955126}{y}}\]
x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}
↓
x + \frac{-1}{x + \frac{e^{z} \cdot -1.1283791670955126}{y}}
(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))
↓
(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (+ x (/ (* (exp z) -1.1283791670955126) y)))))
double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}
↓
double code(double x, double y, double z) {
	return x + (-1.0 / (x + ((exp(z) * -1.1283791670955126) / y)));
}

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

Original2.8
Target0.0
Herbie0.0
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 2.8

    \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
  2. Using strategy rm

  3. Applied clear-num_binary64_120122.8

    \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}}\]

  4. Simplified2.8

    \[\leadsto x + \frac{1}{\color{blue}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}}\]
  5. Using strategy rm

  6. Applied frac-2neg_binary64_120242.8

    \[\leadsto x + \color{blue}{\frac{-1}{-\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}}\]

  7. Simplified2.8

    \[\leadsto x + \frac{\color{blue}{-1}}{-\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}\]

  8. Simplified0.0

    \[\leadsto x + \frac{-1}{\color{blue}{\frac{e^{z} \cdot -1.1283791670955126}{y} + x}}\]

  9. Final simplification0.0

    \[\leadsto x + \frac{-1}{x + \frac{e^{z} \cdot -1.1283791670955126}{y}}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))