Average Error: 60.0 → 3.7

Time: 2.1min

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	60.0
Target	14.9
Herbie	3.7

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

Alternatives

Alternative 1
Error	11.5
Cost	1411

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2920300215033907 \cdot 10^{-90}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;b \leq -3.3352893089214074 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 6.309301555959777 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \end{array}\]

Alternative 2
Error	18.9
Cost	785

\[\begin{array}{l} \mathbf{if}\;a \leq -2471887616062.6514 \lor \neg \left(a \leq -1.2371476513752068 \cdot 10^{-83} \lor \neg \left(a \leq -1.2331491627376912 \cdot 10^{-88}\right) \land a \leq 2.862064489061603 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array}\]

Alternative 3
Error	33.2
Cost	513

\[\begin{array}{l} \mathbf{if}\;a \leq 1.9330188884813744 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 4
Error	60.7
Cost	64

\[0\]

Alternative 5
Error	61.9
Cost	64

\[-1\]

Derivation

Initial program 60.0
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 56.7
\[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Simplified56.5
\[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot \left(a + \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 14.9
\[\leadsto \color{blue}{\frac{a + b}{a \cdot b}}\]
Taylor expanded around 0 3.7
\[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
Simplified3.7
\[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Using strategy rm
Applied *-un-lft-identity_binary64_7603.7
\[\leadsto \frac{1}{a} + \frac{1}{\color{blue}{1 \cdot b}}\]
Applied *-un-lft-identity_binary64_7603.7
\[\leadsto \frac{1}{a} + \frac{\color{blue}{1 \cdot 1}}{1 \cdot b}\]
Applied times-frac_binary64_7663.7
\[\leadsto \frac{1}{a} + \color{blue}{\frac{1}{1} \cdot \frac{1}{b}}\]
Applied *-un-lft-identity_binary64_7603.7
\[\leadsto \frac{1}{\color{blue}{1 \cdot a}} + \frac{1}{1} \cdot \frac{1}{b}\]
Applied *-un-lft-identity_binary64_7603.7
\[\leadsto \frac{\color{blue}{1 \cdot 1}}{1 \cdot a} + \frac{1}{1} \cdot \frac{1}{b}\]
Applied times-frac_binary64_7663.7
\[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{a}} + \frac{1}{1} \cdot \frac{1}{b}\]
Applied distribute-lft-out_binary64_7113.7
\[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{1}{a} + \frac{1}{b}\right)}\]
Simplified3.7
\[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Final simplification3.7
\[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2021014 
(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))))