expq3 (problem 3.4.2)

Average Error: 60.5 → 47.5

Time: 13.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 4.5123883747080296 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{1}{\frac{{\left(e^{a}\right)}^{\varepsilon} - 1}{\frac{{\left(e^{a + b}\right)}^{\varepsilon} - 1}{b + \left(\log 1 + \varepsilon \cdot \left(0.5 \cdot {\left(\log 1\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \end{array}\]

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 (- (exp (* eps (+ a b))) 1.0))
           (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
          (- INFINITY))
         (not
          (<=
           (/
            (* eps (- (exp (* eps (+ a b))) 1.0))
            (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
           4.5123883747080296e-39)))
   (/
    1.0
    (/
     (- (pow (exp a) eps) 1.0)
     (/
      (- (pow (exp (+ a b)) eps) 1.0)
      (+ b (+ (log 1.0) (* eps (* 0.5 (pow (log 1.0) 2.0))))))))
   (/
    (* eps (- (exp (* eps (+ a b))) 1.0))
    (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))))

C

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

↓

double code(double a, double b, double eps) {
	double tmp;
	if ((((((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0))))) <= ((double) -(((double) INFINITY)))) || !((((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0))))) <= 4.5123883747080296e-39))) {
		tmp = (1.0 / (((double) (((double) pow(((double) exp(a)), eps)) - 1.0)) / (((double) (((double) pow(((double) exp(((double) (a + b)))), eps)) - 1.0)) / ((double) (b + ((double) (((double) log(1.0)) + ((double) (eps * ((double) (0.5 * ((double) pow(((double) log(1.0)), 2.0)))))))))))));
	} else {
		tmp = (((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0)))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	14.6
Herbie	47.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < -inf.0 or 4.51238837470802961e-39 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program Error: 63.7 bits

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

   2. Simplified Error: 63.4 bits

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

   3. Taylor expanded around 0 Error: 55.3 bits

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

   4. Simplified Error: 55.3 bits

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

   6. Applied associate-*r/ Error: 55.0 bits

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

   8. Applied clear-num Error: 55.0 bits

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

   9. Simplified Error: 50.0 bits

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 4.51238837470802961e-39

   1. Initial program Error: 3.4 bits

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

4. Final simplification Error: 47.5 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 4.5123883747080296 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{1}{\frac{{\left(e^{a}\right)}^{\varepsilon} - 1}{\frac{{\left(e^{a + b}\right)}^{\varepsilon} - 1}{b + \left(\log 1 + \varepsilon \cdot \left(0.5 \cdot {\left(\log 1\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))