expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 24.8s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

(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)))

C

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

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.0
Herbie	3.4

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

Alternatives

Alternative 1
Error	55.1
Cost	60608

\[\sqrt[3]{\frac{-1 + e^{\varepsilon \cdot \left(b + a\right)}}{b \cdot \left(-1 + e^{\varepsilon \cdot a}\right)}} \cdot \left(\sqrt[3]{\frac{-1 + e^{\varepsilon \cdot \left(b + a\right)}}{b \cdot \left(-1 + e^{\varepsilon \cdot a}\right)}} \cdot \sqrt[3]{\frac{-1 + e^{\varepsilon \cdot \left(b + a\right)}}{b \cdot \left(-1 + e^{\varepsilon \cdot a}\right)}}\right)\]

Alternative 2
Error	60.3
Cost	53312

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left(-1 + e^{\varepsilon \cdot a}\right) \cdot \left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right)\right)}\]

Alternative 3
Error	63.4
Cost	48832

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left({\varepsilon}^{4} \cdot \left(b \cdot \left(0.16666666666666666 \cdot {a}^{3}\right) + \left(a \cdot 0.16666666666666666\right) \cdot {b}^{3}\right) + b \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.5 \cdot \left(a \cdot \left(a \cdot {\varepsilon}^{3}\right)\right)\right)\right) + \left(b \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot {\varepsilon}^{3}\right) + 0.25 \cdot \left(\left(a \cdot a\right) \cdot {\varepsilon}^{4}\right)\right)}\]

Alternative 4
Error	61.7
Cost	47040

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

Alternative 5
Error	61.1
Cost	40576

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left(-1 + e^{\varepsilon \cdot b}\right) \cdot \frac{-1 + {\left(e^{\varepsilon \cdot a}\right)}^{3}}{1 + e^{\varepsilon \cdot a} \cdot \left(e^{\varepsilon \cdot a} + 1\right)}}\]

Alternative 6
Error	61.1
Cost	40576

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left(-1 + e^{\varepsilon \cdot a}\right) \cdot \frac{-1 + {\left(e^{\varepsilon \cdot b}\right)}^{3}}{1 + e^{\varepsilon \cdot b} \cdot \left(e^{\varepsilon \cdot b} + 1\right)}}\]

Alternative 7
Error	59.1
Cost	40384

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

Alternative 8
Error	55.9
Cost	40320

\[\frac{-1 + e^{\varepsilon \cdot \left(b + a\right)}}{b \cdot \left(-1 + {\left(e^{\varepsilon \cdot a}\right)}^{3}\right)} \cdot \left(\left(e^{\varepsilon \cdot a} + 1\right) + e^{\varepsilon \cdot a} \cdot e^{\varepsilon \cdot a}\right)\]

Alternative 9
Error	61.1
Cost	33792

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

Alternative 10
Error	61.2
Cost	33792

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

Alternative 11
Error	60.3
Cost	33344

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

Alternative 12
Error	56.6
Cost	27008

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

Alternative 13
Error	59.1
Cost	26624

\[\sqrt[3]{{\left(\frac{-1 + e^{\varepsilon \cdot \left(b + a\right)}}{b \cdot \left(-1 + e^{\varepsilon \cdot a}\right)}\right)}^{3}}\]

Alternative 14
Error	59.4
Cost	26560

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

Alternative 15
Error	11.7
Cost	22592

\[\sqrt[3]{\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)} \cdot \left(\sqrt[3]{\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)} \cdot \sqrt[3]{\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)}\right)\]

Alternative 16
Error	60.3
Cost	20544

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

Alternative 17
Error	61.3
Cost	20416

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

Alternative 18
Error	61.2
Cost	20416

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

Alternative 19
Error	57.3
Cost	20352

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

Alternative 20
Error	63.2
Cost	15104

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{a \cdot \left(b \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + {\varepsilon}^{3} \cdot \left(b \cdot \left(0.5 \cdot \left(a \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(a \cdot 0.5\right)\right)}\]

Alternative 21
Error	57.0
Cost	14528

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left(-1 + e^{\varepsilon \cdot b}\right) \cdot \left(\varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right)}\]

Alternative 22
Error	57.0
Cost	14016

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

Alternative 23
Error	57.1
Cost	14016

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

Alternative 24
Error	57.1
Cost	14016

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

Alternative 25
Error	56.9
Cost	14016

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

Alternative 26
Error	62.3
Cost	14016

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

Alternative 27
Error	54.9
Cost	13888

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

Alternative 28
Error	19.2
Cost	13888

\[\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \log \left(e^{\frac{\varepsilon \cdot b}{a}}\right)\right)\right)\]

Alternative 29
Error	61.2
Cost	8000

\[\frac{\varepsilon \cdot \left(-1 + e^{\varepsilon \cdot \left(b + a\right)}\right)}{\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right)}\]

Alternative 30
Error	61.4
Cost	7488

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

Alternative 31
Error	62.2
Cost	7488

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

Alternative 32
Error	26.8
Cost	3008

\[\frac{\left(\frac{1}{b} - 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right) \cdot \left(1 + a \cdot \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)\right)}{a \cdot \left(\frac{1}{b} - 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)}\]

Alternative 33
Error	31.4
Cost	2240

\[\frac{1}{a} + \frac{\left(\varepsilon - \frac{\varepsilon \cdot b}{a}\right) \cdot \left(1 + \left(b \cdot 0.5\right) \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)}{b \cdot \left(\varepsilon - \frac{\varepsilon \cdot b}{a}\right)}\]

Alternative 34
Error	10.6
Cost	1088

\[\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{\varepsilon \cdot b}{a}\right)\right)\]

Alternative 35
Error	10.2
Cost	1088

\[\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + b \cdot \frac{\varepsilon}{a}\right)\right)\]

Alternative 36
Error	3.5
Cost	704

\[\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot 0.5\right)\]

Alternative 37
Error	40.5
Cost	576

\[\frac{1 + b \cdot \left(\varepsilon \cdot 0.5\right)}{a}\]

Alternative 38
Error	15.0
Cost	448

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

Alternative 39
Error	33.1
Cost	192

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

Alternative 40
Error	33.3
Cost	192

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

Alternative 41
Error	61.9
Cost	64

\[1\]

Alternative 42
Error	60.9
Cost	64

\[0\]

Alternative 43
Error	61.9
Cost	64

\[-1\]

Derivation

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 56.9

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

3. Simplified 56.9

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

4. Taylor expanded around 0 10.6

   \[\leadsto \color{blue}{\frac{1}{a} + \left(\frac{1}{b} + \left(0.5 \cdot \frac{\varepsilon \cdot b}{a} + 0.5 \cdot \varepsilon\right)\right)}\]

5. Simplified 10.6

   \[\leadsto \color{blue}{\frac{1}{a} + \left(\frac{1}{b} + 0.5 \cdot \left(\varepsilon + \frac{b \cdot \varepsilon}{a}\right)\right)}\]

6. Taylor expanded around 0 3.4

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

7. Simplified 3.4

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

8. Simplified 3.4

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

9. Final simplification 3.4

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

Reproduce

herbie shell --seed 2021042 
(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))))