Falkner and Boettcher, Appendix A

Average Error: 5.3 → 2.1

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \left(1 + k \cdot 10\right) + k \cdot k\\ \mathbf{if}\;\frac{t_0}{t_1} \leq 8.396832676077389 \cdot 10^{+155}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* a (pow k m))) (t_1 (+ (+ 1.0 (* k 10.0)) (* k k))))
   (if (<= (/ t_0 t_1) 8.396832676077389e+155)
     (/ (* (pow k m) (/ a (sqrt (fma k (+ k 10.0) 1.0)))) (sqrt t_1))
     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 = a * pow(k, m);
	double t_1 = (1.0 + (k * 10.0)) + (k * k);
	double tmp;
	if ((t_0 / t_1) <= 8.396832676077389e+155) {
		tmp = (pow(k, m) * (a / sqrt(fma(k, (k + 10.0), 1.0)))) / sqrt(t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 8.39683267607738909e155

   1. Initial program 1.8

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

   2. Applied add-sqr-sqrt_binary64 1.9

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

   3. Applied associate-/r*_binary64 1.9

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

   4. Simplified 2.3

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

   if 8.39683267607738909e155 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 19.9

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

   2. Simplified 19.8

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

   3. Taylor expanded in k around 0 18.4

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]

   4. Simplified 1.4

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 8.396832676077389 \cdot 10^{+155}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{\left(1 + k \cdot 10\right) + k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))