Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.5

Time: 6.1s

Precision: binary64

Math

\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

↓

\[\begin{array}{l} t_0 := \left(1 + k \cdot 10\right) + k \cdot k\\ t_1 := \frac{a \cdot {k}^{m}}{t_0}\\ \mathbf{if}\;t_1 \leq -1.9188946875267454 \cdot 10^{-291}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{a}{\frac{k}{{k}^{m}}}}{k + 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + a \cdot \left(m \cdot \log k\right)}{t_0}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) (t_1 (/ (* a (pow k m)) t_0)))
   (if (<= t_1 -1.9188946875267454e-291)
     (/ a (/ (fma k (+ k 10.0) 1.0) (pow k m)))
     (if (<= t_1 0.0)
       (/ (/ a (/ k (pow k m))) (+ k 10.0))
       (/ (+ a (* a (* m (log k)))) t_0)))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

↓

double code(double a, double k, double m) {
	double t_0 = (1.0 + (k * 10.0)) + (k * k);
	double t_1 = (a * pow(k, m)) / t_0;
	double tmp;
	if (t_1 <= -1.9188946875267454e-291) {
		tmp = a / (fma(k, (k + 10.0), 1.0) / pow(k, m));
	} else if (t_1 <= 0.0) {
		tmp = (a / (k / pow(k, m))) / (k + 10.0);
	} else {
		tmp = (a + (a * (m * log(k)))) / t_0;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < -1.91889468752674539e-291

   1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Using strategy rm

   3. Applied associate-/l*_binary64 0.1

      \[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]

   4. Simplified 0.1

      \[\leadsto \frac{a}{\color{blue}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}} \]

   if -1.91889468752674539e-291 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < -0.0

   1. Initial program 2.7

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in k around inf 3.0

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{10 \cdot k + {k}^{2}}} \]

   3. Simplified 6.2

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{k}^{m}}{k + 10}} \]
   4. Using strategy rm

   5. Applied associate-*r/_binary64 6.2

      \[\leadsto \color{blue}{\frac{\frac{a}{k} \cdot {k}^{m}}{k + 10}} \]

   6. Simplified 0.3

      \[\leadsto \frac{\color{blue}{\frac{a \cdot {k}^{m}}{k}}}{k + 10} \]
   7. Using strategy rm

   8. Applied associate-/l*_binary64 0.3

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{k}{{k}^{m}}}}}{k + 10} \]

   if -0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 0.3

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0 1.4

      \[\leadsto \frac{\color{blue}{a \cdot \left(\log k \cdot m\right) + a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq -1.9188946875267454 \cdot 10^{-291}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{\frac{a}{\frac{k}{{k}^{m}}}}{k + 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + a \cdot \left(m \cdot \log k\right)}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))