expq3 (problem 3.4.2)

Average Error: 60.0 → 0.5

Time: 24.3s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.914699664673985 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \log \left(e^{e^{\varepsilon \cdot b} - 1}\right)}\\ \end{array}\]

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)))
           1.914699664673985e-31)))
   (+ (/ 1.0 a) (/ 1.0 b))
   (/
    (* eps (- (exp (* eps (+ a b))) 1.0))
    (* (- (exp (* eps a)) 1.0) (log (exp (- (exp (* eps b)) 1.0)))))))

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)) || !(((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= 1.914699664673985e-31)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * log(exp(exp(eps * b) - 1.0)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.0
Target	15.2
Herbie	0.5

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

Derivation

1. Split input into 2 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 1.9146996646739851e-31 < (/.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 63.7

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

   2. Taylor expanded around 0 58.0

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

   3. Simplified 57.7

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

   4. Taylor expanded around 0 13.8

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

   5. Simplified 13.8

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

   6. Taylor expanded around 0 0.3

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

   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))) < 1.9146996646739851e-31

   1. Initial program 3.3

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

   3. Applied add-log-exp_binary64_1481 3.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.914699664673985 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \log \left(e^{e^{\varepsilon \cdot b} - 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021027 
(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))))