Average Error: 60.4 → 3.3
Time: 29.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\frac{1}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r94190 = eps;
        double r94191 = a;
        double r94192 = b;
        double r94193 = r94191 + r94192;
        double r94194 = r94193 * r94190;
        double r94195 = exp(r94194);
        double r94196 = 1.0;
        double r94197 = r94195 - r94196;
        double r94198 = r94190 * r94197;
        double r94199 = r94191 * r94190;
        double r94200 = exp(r94199);
        double r94201 = r94200 - r94196;
        double r94202 = r94192 * r94190;
        double r94203 = exp(r94202);
        double r94204 = r94203 - r94196;
        double r94205 = r94201 * r94204;
        double r94206 = r94198 / r94205;
        return r94206;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r94207 = 1.0;
        double r94208 = b;
        double r94209 = r94207 / r94208;
        double r94210 = a;
        double r94211 = r94207 / r94210;
        double r94212 = r94209 + r94211;
        return r94212;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.4
Target14.6
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  2. Taylor expanded around 0 3.3

    \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]

  3. Final simplification3.3

    \[\leadsto \frac{1}{b} + \frac{1}{a}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))