Average Error: 58.4 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon}{\frac{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}} \le -21560263070086.36:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon}{\frac{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}} \le 7.820264484588096 \cdot 10^{-45}:\\ \;\;\;\;\frac{\varepsilon}{\frac{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.6
Herbie0.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ eps (/ (* (expm1 (* b eps)) (expm1 (* eps a))) (expm1 (* (+ b a) eps)))) < -21560263070086.36 or 7.820264484588096e-45 < (/ eps (/ (* (expm1 (* b eps)) (expm1 (* eps a))) (expm1 (* (+ b a) eps))))

    1. Initial program 61.6

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

    2. Taylor expanded around 0 0.5

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

    if -21560263070086.36 < (/ eps (/ (* (expm1 (* b eps)) (expm1 (* eps a))) (expm1 (* (+ b a) eps)))) < 7.820264484588096e-45

    1. Initial program 42.6

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

    3. Applied associate-/l*42.6

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

    4. Applied simplify0.5

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +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))))