Average Error: 58.6 → 1.0
Time: 3.0m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right) \le -808079.2307463246:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right) \le 2.7580528090683865 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.2
Herbie1.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (/ eps (expm1 (fma (+ b b) eps (* eps b)))) (/ (expm1 (* a eps)) (expm1 (* (+ a b) eps)))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))) < -808079.2307463246 or 2.7580528090683865e-18 < (* (/ (/ eps (expm1 (fma (+ b b) eps (* eps b)))) (/ (expm1 (* a eps)) (expm1 (* (+ a b) eps)))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))

    1. Initial program 60.9

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

    2. Taylor expanded around 0 1.1

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

    if -808079.2307463246 < (* (/ (/ eps (expm1 (fma (+ b b) eps (* eps b)))) (/ (expm1 (* a eps)) (expm1 (* (+ a b) eps)))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))) < 2.7580528090683865e-18

    1. Initial program 47.8

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

    3. Applied flip3--47.8

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

    4. Applied associate-*r/47.8

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

    5. Applied associate-/r/47.8

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

    6. Applied simplify0.3

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

  4. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right) \le -808079.2307463246:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right) \le 2.7580528090683865 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\varepsilon}{(e^{(\left(b + b\right) \cdot \varepsilon + \left(b \cdot \varepsilon\right))_*} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}} \cdot \left(\left(1 + e^{b \cdot \varepsilon}\right) + e^{b \cdot \varepsilon} \cdot e^{b \cdot \varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(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))))