Average Error: 60.3 → 3.4

Time: 1.2min

Precision: binary64

Cost: 448

\[-1 < \varepsilon \land \varepsilon < 1\]

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

↓

\[\frac{1}{a} + \frac{1}{b}\]

\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)}

↓

\frac{1}{a} + \frac{1}{b}

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

↓

(FPCore (a b eps) :precision binary64 (+ (/ 1.0 a) (/ 1.0 b)))

double code(double a, double b, double eps) {
	return (eps * (exp((a + b) * eps) - 1.0)) / ((exp(a * eps) - 1.0) * (exp(b * eps) - 1.0));
}

↓

double code(double a, double b, double eps) {
	return (1.0 / a) + (1.0 / b);
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	60.3
Target	14.8
Herbie	3.4

\[\frac{a + b}{a \cdot b}\]

Alternatives

Alternative 1
Error	11.1
Cost	1411

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1051454215971253 \cdot 10^{-119}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;b \leq 1.2536815477017208 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 6.309638141211083 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \end{array}\]

Alternative 2
Error	19.0
Cost	785

\[\begin{array}{l} \mathbf{if}\;a \leq -7.675712136404333 \cdot 10^{-111} \lor \neg \left(a \leq 3.807625924873168 \cdot 10^{-58} \lor \neg \left(a \leq 2.4405151995062135 \cdot 10^{+24}\right) \land a \leq 1.6276863144420978 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array}\]

Alternative 3
Error	33.1
Cost	192

\[\frac{1}{b}\]

Alternative 4
Error	60.9
Cost	64

\[0\]

Alternative 5
Error	61.9
Cost	64

\[1\]

Derivation

Initial program 60.3
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 62.2
\[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{a \cdot \left({\varepsilon}^{2} \cdot b\right)}}\]
Simplified62.2
\[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{a \cdot \left(b \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}\]
Taylor expanded around 0 14.8
\[\leadsto \color{blue}{\frac{a + b}{a \cdot b}}\]
Taylor expanded around 0 3.4
\[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
Simplified3.4
\[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Using strategy rm
Applied *-un-lft-identity_binary64_7603.4
\[\leadsto \frac{1}{a} + \frac{1}{\color{blue}{1 \cdot b}}\]
Applied add-sqr-sqrt_binary64_7823.4
\[\leadsto \frac{1}{a} + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot b}\]
Applied times-frac_binary64_7663.4
\[\leadsto \frac{1}{a} + \color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}}\]
Applied *-un-lft-identity_binary64_7603.4
\[\leadsto \frac{1}{\color{blue}{1 \cdot a}} + \frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}\]
Applied add-sqr-sqrt_binary64_7823.4
\[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot a} + \frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}\]
Applied times-frac_binary64_7663.4
\[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{a}} + \frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}\]
Applied distribute-lft-out_binary64_7113.4
\[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{a} + \frac{\sqrt{1}}{b}\right)}\]
Simplified3.4
\[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\left(\frac{1}{a} + \frac{1}{b}\right)}\]
Simplified3.4
\[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Final simplification3.4
\[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))