Average Error: 58.6 → 0.4
Time: 37.4s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*} \le -0.2684139380331385:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*} \le 2.0738728361243063 \cdot 10^{-50}:\\ \;\;\;\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target13.8
Herbie0.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (expm1 (* (+ a b) eps)) (expm1 (* a eps))) (/ eps (expm1 (* b eps)))) < -0.2684139380331385 or 2.0738728361243063e-50 < (* (/ (expm1 (* (+ a b) eps)) (expm1 (* a eps))) (/ eps (expm1 (* b eps))))

    1. Initial program 61.7

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

    2. Applied simplify33.1

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

    3. Taylor expanded around 0 0.4

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

    if -0.2684139380331385 < (* (/ (expm1 (* (+ a b) eps)) (expm1 (* a eps))) (/ eps (expm1 (* b eps)))) < 2.0738728361243063e-50

    1. Initial program 43.7

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

    2. Applied simplify0.3

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

    4. Applied div-inv0.4

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

    5. Applied div-inv0.4

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

    6. Applied times-frac0.4

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

    7. Applied simplify0.3

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

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))