expq3 (problem 3.4.2)

Average Error: 60.7 → 0.9

Time: 12.4s

Precision: binary64

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

[a b]: =sort([a b])

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.2960983203071421 \cdot 10^{-145}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \log \left(e^{e^{\varepsilon \cdot a} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

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))
   (+ (/ 1.0 b) (+ (/ 1.0 a) (* eps 0.5)))
   (if (<=
        (/
         (* eps (- (exp (* eps (+ a b))) 1.0))
         (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
        1.2960983203071421e-145)
     (/
      (* eps (- (exp (* eps (+ a b))) 1.0))
      (* (- (exp (* eps b)) 1.0) (log (exp (- (exp (* eps a)) 1.0)))))
     (+ (/ 1.0 b) (/ 1.0 a)))))

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 = (1.0 / b) + ((1.0 / a) + (eps * 0.5));
	} else if (((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= 1.2960983203071421e-145) {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * b) - 1.0) * log(exp(exp(eps * a) - 1.0)));
	} else {
		tmp = (1.0 / b) + (1.0 / a);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.7
Target	14.6
Herbie	0.9

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

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

   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 22.9

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

   3. Simplified 22.9

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

   4. Taylor expanded around 0 3.2

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

   5. Simplified 3.2

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

   6. Taylor expanded around 0 0.0

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

   7. Simplified 0.0

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

   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))) < 1.29609832030714214e-145

   1. Initial program 3.8

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

   3. Applied add-log-exp_binary64_1481 3.8

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

   4. Applied add-log-exp_binary64_1481 4.1

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

   5. Applied diff-log_binary64_1534 4.2

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

   6. Simplified 4.1

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

   if 1.29609832030714214e-145 < (/.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 63.1

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

   2. Taylor expanded around 0 63.2

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

   3. Simplified 63.2

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

   4. Taylor expanded around 0 9.2

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

   5. Simplified 9.2

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

   6. Taylor expanded around 0 0.8

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

   7. Simplified 0.8

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.2960983203071421 \cdot 10^{-145}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \log \left(e^{e^{\varepsilon \cdot a} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))))