Average Error: 58.7 → 3.3
Time: 1.7m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \varepsilon \cdot b\right)\right)} \le -3.940853444483336 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \varepsilon \cdot b\right)\right)} \le 4.1454636629698146 \cdot 10^{-73}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(\sqrt{e^{b \cdot \varepsilon}} + 1\right) \cdot \left(\sqrt{e^{b \cdot \varepsilon}} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.7
Target14.1
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

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))))) < -3.940853444483336e+24 or 4.1454636629698146e-73 < (/ (* 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 59.7

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

    2. Taylor expanded around 0 2.3

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

    if -3.940853444483336e+24 < (/ (* 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))))) < 4.1454636629698146e-73

    1. Initial program 30.4

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

    3. Applied add-sqr-sqrt30.5

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

    4. Applied difference-of-sqr-130.5

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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))))