Average Error: 58.4 → 1.8
Time: 3.9m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right) \le -3.0522411179596246 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right) \le 1.850491661493059 \cdot 10^{-67}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target13.9
Herbie1.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (expm1 (* (+ a b) eps)) (* (expm1 (fma b (+ eps eps) (* eps b))) (/ (expm1 (* a eps)) eps))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))) < -3.0522411179596246e-34 or 1.850491661493059e-67 < (* (/ (expm1 (* (+ a b) eps)) (* (expm1 (fma b (+ eps eps) (* eps b))) (/ (expm1 (* a eps)) eps))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))

    1. Initial program 60.2

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

    2. Taylor expanded around 0 1.9

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

    if -3.0522411179596246e-34 < (* (/ (expm1 (* (+ a b) eps)) (* (expm1 (fma b (+ eps eps) (* eps b))) (/ (expm1 (* a eps)) eps))) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))) < 1.850491661493059e-67

    1. Initial program 44.9

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

    3. Applied flip3--45.1

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

    4. Applied associate-*r/45.1

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

    5. Applied associate-/r/45.1

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

    6. Applied simplify0.7

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

  4. Applied simplify1.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right) \le -3.0522411179596246 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right) \le 1.850491661493059 \cdot 10^{-67}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{(b \cdot \left(\varepsilon + \varepsilon\right) + \left(\varepsilon \cdot b\right))_*} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{\varepsilon}} \cdot \left(\left(1 + e^{\varepsilon \cdot b}\right) + e^{\varepsilon \cdot b} \cdot e^{\varepsilon \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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))))