expq3 (problem 3.4.2)

Average Error: 60.1 → 4.1

Time: 10.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.009444801518506 \cdot 10^{-57} \lor \neg \left(\varepsilon \leq -2.010446323295612 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \log \left(e^{-1 + e^{\varepsilon \cdot b}}\right)}\\ \end{array}\]

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
 (if (or (<= eps -9.009444801518506e-57) (not (<= eps -2.010446323295612e-63)))
   (+ (/ 1.0 b) (/ 1.0 a))
   (/
    (* eps (- (exp (* eps (+ b a))) 1.0))
    (* (- (exp (* eps a)) 1.0) (log (exp (+ -1.0 (exp (* eps 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) {
	double tmp;
	if ((eps <= -9.009444801518506e-57) || !(eps <= -2.010446323295612e-63)) {
		tmp = (1.0 / b) + (1.0 / a);
	} else {
		tmp = (eps * (exp(eps * (b + a)) - 1.0)) / ((exp(eps * a) - 1.0) * log(exp(-1.0 + exp(eps * b))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.1
Target	14.8
Herbie	4.1

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

Derivation

1. Split input into 2 regimes

2. if eps < -9.00944480151850616e-57 or -2.010446323295612e-63 < eps

   1. Initial program 60.1

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

   2. Taylor expanded around 0 57.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.16666666666666666 \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   3. Simplified 57.7

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

   4. Taylor expanded around 0 56.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   5. Simplified 56.4

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

   6. Taylor expanded around 0 3.5

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

   if -9.00944480151850616e-57 < eps < -2.010446323295612e-63

   1. Initial program 57.9

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

   3. Applied add-log-exp_binary64 57.9

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

   4. Applied add-log-exp_binary64 57.9

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

   5. Applied diff-log_binary64 57.9

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

   6. Simplified 57.9

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.009444801518506 \cdot 10^{-57} \lor \neg \left(\varepsilon \leq -2.010446323295612 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \log \left(e^{-1 + e^{\varepsilon \cdot b}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))))