Average Error: 58.6 → 3.1
Time: 1.1m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}}\right) \le -3.30870715340429 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} + \frac{1}{a}}}\right) \le 1.2837198413347267 \cdot 10^{-159}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(\sqrt{e^{a \cdot \varepsilon}} + 1\right) \cdot \left(\sqrt{e^{a \cdot \varepsilon}} - 1\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.4
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (* (* (cbrt (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a))))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a)))))) < -3.30870715340429e-121 or 1.2837198413347267e-159 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (* (* (cbrt (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a))))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a))))))

    1. Initial program 60.3

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

    2. Taylor expanded around 0 1.8

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

    if -3.30870715340429e-121 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (* (* (cbrt (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a))))) (cbrt (cbrt (+ (/ 1 b) (/ 1 a)))))) < 1.2837198413347267e-159

    1. Initial program 28.0

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

    3. Applied add-sqr-sqrt28.1

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

    4. Applied difference-of-sqr-128.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))))