(exp (/ (* (- (- x u)) (- x u)) (* (* 2 s) s)))

Average Error: 7.9 → 4.3

Time: 1.4s

Precision: binary64

Math

\[e^{\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s} \le -4.01093996882167036 \cdot 10^{-243}:\\ \;\;\;\;e^{\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{u - x}\right)}^{\left(\frac{x - u}{\left(2 \cdot s\right) \cdot s}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus u

Bits error versus s

Derivation

1. Split input into 2 regimes

2. if (/ (* (neg (- x u)) (- x u)) (* (* 2.0 s) s)) < -4.01093996882167036e-243

   1. Initial program 0.0

      \[e^{\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s}}\]

   if -4.01093996882167036e-243 < (/ (* (neg (- x u)) (- x u)) (* (* 2.0 s) s))

   1. Initial program 24.3

      \[e^{\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s}}\]

   2. Simplified 13.4

      \[\leadsto \color{blue}{{\left(e^{u - x}\right)}^{\left(\frac{x - u}{\left(2 \cdot s\right) \cdot s}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s} \le -4.01093996882167036 \cdot 10^{-243}:\\ \;\;\;\;e^{\frac{\left(-\left(x - u\right)\right) \cdot \left(x - u\right)}{\left(2 \cdot s\right) \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{u - x}\right)}^{\left(\frac{x - u}{\left(2 \cdot s\right) \cdot s}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x u s)
  :name "(exp (/ (* (- (- x u)) (- x u)) (* (* 2 s) s)))"
  :precision binary64
  (exp (/ (* (neg (- x u)) (- x u)) (* (* 2.0 s) s))))