Average Error: 59.0 → 3.0
Time: 2.7m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}}{(e^{\varepsilon \cdot b} - 1)^*} \le -3.714349213308675 \cdot 10^{+273}:\\ \;\;\;\;(\left(\frac{\varepsilon}{b}\right) \cdot a + \left(\frac{1}{a}\right))_* + \frac{1}{b}\\ \mathbf{if}\;\frac{\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}}{(e^{\varepsilon \cdot b} - 1)^*} \le 4.2539049380951326 \cdot 10^{+222}:\\ \;\;\;\;\frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\varepsilon}{b}\right) \cdot a + \left(\frac{1}{a}\right))_* + \frac{1}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original59.0
Target14.3
Herbie3.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (expm1 (* (+ a b) eps)) (/ (expm1 (* a eps)) eps)) (expm1 (* eps b))) < -3.714349213308675e+273 or 4.2539049380951326e+222 < (/ (/ (expm1 (* (+ a b) eps)) (/ (expm1 (* a eps)) eps)) (expm1 (* eps b)))

    1. Initial program 61.8

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

    2. Taylor expanded around 0 4.4

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

    3. Applied simplify4.4

      \[\leadsto \color{blue}{(\left(\frac{\varepsilon}{b}\right) \cdot a + \left(\frac{1}{a}\right))_* + \frac{1}{b}}\]

    if -3.714349213308675e+273 < (/ (/ (expm1 (* (+ a b) eps)) (/ (expm1 (* a eps)) eps)) (expm1 (* eps b))) < 4.2539049380951326e+222

    1. Initial program 56.6

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

    3. Applied times-frac56.6

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

    4. Applied simplify45.7

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

    5. Applied simplify1.8

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018195 +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))))