expq3 (problem 3.4.2)

Average Error: 60.2 → 2.3

Time: 15.4s

Precision: binary64

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

\[[a, b]=\mathsf{sort}([a, b])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 0.9566692163658991:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\log \left({\left(e^{e^{\varepsilon \cdot b} - 1}\right)}^{\left(e^{\varepsilon \cdot a} - 1\right)}\right)}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq \infty:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \left(0.16666666666666666 \cdot \frac{a \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{b} + \left(\left(\varepsilon \cdot \left(\varepsilon \cdot b\right)\right) \cdot 0.08333333333333333 + \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.25\right)\right)\right)\\ \end{array}\]

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 (<=
      (/
       (* eps (- (exp (* eps (+ a b))) 1.0))
       (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
      (- INFINITY))
   (/ (+ a b) (* a b))
   (if (<=
        (/
         (* eps (- (exp (* eps (+ a b))) 1.0))
         (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
        0.9566692163658991)
     (/
      (* eps (- (exp (* eps (+ a b))) 1.0))
      (log (pow (exp (- (exp (* eps b)) 1.0)) (- (exp (* eps a)) 1.0))))
     (if (<=
          (/
           (* eps (- (exp (* eps (+ a b))) 1.0))
           (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
          INFINITY)
       (/ (+ a b) (* a b))
       (+
        (/ 1.0 a)
        (+
         (/ 1.0 b)
         (+
          (* 0.16666666666666666 (/ (* a (* a (* eps eps))) b))
          (+
           (* (* eps (* eps b)) 0.08333333333333333)
           (* (* a (* eps eps)) 0.25)))))))))

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 * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= -((double) INFINITY)) {
		tmp = (a + b) / (a * b);
	} else if (((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= 0.9566692163658991) {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / log(pow(exp(exp(eps * b) - 1.0), (exp(eps * a) - 1.0)));
	} else if (((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= ((double) INFINITY)) {
		tmp = (a + b) / (a * b);
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + ((0.16666666666666666 * ((a * (a * (eps * eps))) / b)) + (((eps * (eps * b)) * 0.08333333333333333) + ((a * (eps * eps)) * 0.25))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	15.0
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.0 or 0.956669216365899078 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < +inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around 0 40.5

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

   3. Simplified 40.5

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

   4. Taylor expanded around 0 7.2

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

   if -inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 0.956669216365899078

   1. Initial program 3.9

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

   3. Applied add-log-exp_binary64_1140 5.1

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

   4. Simplified 5.1

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

   if +inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

   1. Initial program 64.0

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

   2. Taylor expanded around 0 64.0

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

   3. Simplified 64.0

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

   4. Taylor expanded around 0 5.6

      \[\leadsto \color{blue}{\frac{1}{a} + \left(\frac{1}{b} + \left(0.16666666666666666 \cdot \frac{{a}^{2} \cdot {\varepsilon}^{2}}{b} + \left(0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right) + 0.25 \cdot \left(a \cdot {\varepsilon}^{2}\right)\right)\right)\right)}\]

   5. Simplified 5.6

      \[\leadsto \color{blue}{\frac{1}{a} + \left(\frac{1}{b} + \left(0.16666666666666666 \cdot \frac{\left(a \cdot a\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}{b} + \left(\left(\varepsilon \cdot \left(\varepsilon \cdot b\right)\right) \cdot 0.08333333333333333 + 0.25 \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*l*_binary64_1042 0.2

      \[\leadsto \frac{1}{a} + \left(\frac{1}{b} + \left(0.16666666666666666 \cdot \frac{\color{blue}{a \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{b} + \left(\left(\varepsilon \cdot \left(\varepsilon \cdot b\right)\right) \cdot 0.08333333333333333 + 0.25 \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 0.9566692163658991:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\log \left({\left(e^{e^{\varepsilon \cdot b} - 1}\right)}^{\left(e^{\varepsilon \cdot a} - 1\right)}\right)}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq \infty:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \left(0.16666666666666666 \cdot \frac{a \cdot \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{b} + \left(\left(\varepsilon \cdot \left(\varepsilon \cdot b\right)\right) \cdot 0.08333333333333333 + \left(a \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.25\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))))