Average Error: 60.3 → 54.8
Time: 11.5s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.949192232894134483079179646343452700776 \cdot 10^{49}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le 3.146231950313864865160087198758186099744 \cdot 10^{60}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.949192232894134483079179646343452700776 \cdot 10^{49}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\

\mathbf{elif}\;a \le 3.146231950313864865160087198758186099744 \cdot 10^{60}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\

\end{array}
double f(double a, double b, double eps) {
        double r97689 = eps;
        double r97690 = a;
        double r97691 = b;
        double r97692 = r97690 + r97691;
        double r97693 = r97692 * r97689;
        double r97694 = exp(r97693);
        double r97695 = 1.0;
        double r97696 = r97694 - r97695;
        double r97697 = r97689 * r97696;
        double r97698 = r97690 * r97689;
        double r97699 = exp(r97698);
        double r97700 = r97699 - r97695;
        double r97701 = r97691 * r97689;
        double r97702 = exp(r97701);
        double r97703 = r97702 - r97695;
        double r97704 = r97700 * r97703;
        double r97705 = r97697 / r97704;
        return r97705;
}
↓
double f(double a, double b, double eps) {
        double r97706 = a;
        double r97707 = -1.9491922328941345e+49;
        bool r97708 = r97706 <= r97707;
        double r97709 = eps;
        double r97710 = b;
        double r97711 = r97706 + r97710;
        double r97712 = r97711 * r97709;
        double r97713 = exp(r97712);
        double r97714 = 1.0;
        double r97715 = r97713 - r97714;
        double r97716 = r97709 * r97715;
        double r97717 = r97706 * r97709;
        double r97718 = exp(r97717);
        double r97719 = r97718 - r97714;
        double r97720 = 0.16666666666666666;
        double r97721 = 3.0;
        double r97722 = pow(r97709, r97721);
        double r97723 = pow(r97710, r97721);
        double r97724 = r97722 * r97723;
        double r97725 = 0.5;
        double r97726 = 2.0;
        double r97727 = pow(r97709, r97726);
        double r97728 = pow(r97710, r97726);
        double r97729 = r97727 * r97728;
        double r97730 = r97709 * r97710;
        double r97731 = fma(r97725, r97729, r97730);
        double r97732 = fma(r97720, r97724, r97731);
        double r97733 = r97719 * r97732;
        double r97734 = r97716 / r97733;
        double r97735 = 3.146231950313865e+60;
        bool r97736 = r97706 <= r97735;
        double r97737 = pow(r97706, r97721);
        double r97738 = r97737 * r97722;
        double r97739 = pow(r97706, r97726);
        double r97740 = r97739 * r97727;
        double r97741 = fma(r97725, r97740, r97717);
        double r97742 = fma(r97720, r97738, r97741);
        double r97743 = exp(r97730);
        double r97744 = r97743 - r97714;
        double r97745 = r97742 * r97744;
        double r97746 = r97716 / r97745;
        double r97747 = r97719 * r97744;
        double r97748 = r97716 / r97747;
        double r97749 = r97736 ? r97746 : r97748;
        double r97750 = r97708 ? r97734 : r97749;
        return r97750;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original60.3
Target14.7
Herbie54.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.9491922328941345e+49

    1. Initial program 54.7

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

    2. Taylor expanded around 0 48.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

    3. Simplified48.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}}\]

    if -1.9491922328941345e+49 < a < 3.146231950313865e+60

    1. Initial program 63.7

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

    2. Taylor expanded around inf 63.7

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

    3. Taylor expanded around 0 56.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\]

    4. Simplified56.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right)} \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\]

    if 3.146231950313865e+60 < a

    1. Initial program 53.9

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

    2. Taylor expanded around inf 53.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(e^{\varepsilon \cdot b} - 1\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification54.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.949192232894134483079179646343452700776 \cdot 10^{49}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le 3.146231950313864865160087198758186099744 \cdot 10^{60}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))))