Average Error: 58.4 → 2.9
Time: 59.8s
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \varepsilon \cdot b\right)\right)} \le -2.3644604257159884 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \varepsilon \cdot b\right)\right)} \le 6.96886894787775 \cdot 10^{-47}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(\sqrt{e^{a \cdot \varepsilon}} + 1\right) \cdot \left(\sqrt{e^{a \cdot \varepsilon}} - 1\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.2
Herbie2.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (+ (* 1/6 (* (pow eps 3) (pow b 3))) (* eps b))))) < -2.3644604257159884e-72 or 6.96886894787775e-47 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (+ (* 1/6 (* (pow eps 3) (pow b 3))) (* eps b)))))

    1. Initial program 59.6

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

    2. Taylor expanded around 0 2.4

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

    if -2.3644604257159884e-72 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (+ (* 1/6 (* (pow eps 3) (pow b 3))) (* eps b))))) < 6.96886894787775e-47

    1. Initial program 17.7

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt17.8

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

    4. Applied difference-of-sqr-117.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\left(\sqrt{e^{a \cdot \varepsilon}} + 1\right) \cdot \left(\sqrt{e^{a \cdot \varepsilon}} - 1\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 59.8s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))