NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 1.0

Time: 9.7s

Precision: binary64

Math

\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 244.84131261463813:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}} \cdot \sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}} \cdot \frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \frac{\sqrt[3]{1 + \frac{1}{\varepsilon}}}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x 244.84131261463813)
   (/
    (*
     (/
      (*
       (cbrt (+ (* (pow x 9.0) 0.2962962962962963) (pow (- 2.0 (* x x)) 3.0)))
       (cbrt (+ (* (pow x 9.0) 0.2962962962962963) (pow (- 2.0 (* x x)) 3.0))))
      (sqrt
       (+
        (* (pow x 6.0) 0.4444444444444444)
        (*
         (- 2.0 (* x x))
         (- (- 2.0 (* x x)) (* (pow x 3.0) 0.6666666666666666))))))
     (/
      (cbrt (+ (* (pow x 9.0) 0.2962962962962963) (pow (- 2.0 (* x x)) 3.0)))
      (sqrt
       (+
        (* (pow x 6.0) 0.4444444444444444)
        (*
         (- 2.0 (* x x))
         (- (- 2.0 (* x x)) (* (pow x 3.0) 0.6666666666666666)))))))
    2.0)
   (/
    (-
     (*
      (* (cbrt (+ 1.0 (/ 1.0 eps))) (cbrt (+ 1.0 (/ 1.0 eps))))
      (/ (cbrt (+ 1.0 (/ 1.0 eps))) (exp (* x (- 1.0 eps)))))
     (* (- (/ 1.0 eps) 1.0) (exp (* x (- -1.0 eps)))))
    2.0)))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

↓

double code(double x, double eps) {
	double tmp;
	if (x <= 244.84131261463813) {
		tmp = (((cbrt((pow(x, 9.0) * 0.2962962962962963) + pow((2.0 - (x * x)), 3.0)) * cbrt((pow(x, 9.0) * 0.2962962962962963) + pow((2.0 - (x * x)), 3.0))) / sqrt((pow(x, 6.0) * 0.4444444444444444) + ((2.0 - (x * x)) * ((2.0 - (x * x)) - (pow(x, 3.0) * 0.6666666666666666))))) * (cbrt((pow(x, 9.0) * 0.2962962962962963) + pow((2.0 - (x * x)), 3.0)) / sqrt((pow(x, 6.0) * 0.4444444444444444) + ((2.0 - (x * x)) * ((2.0 - (x * x)) - (pow(x, 3.0) * 0.6666666666666666)))))) / 2.0;
	} else {
		tmp = (((cbrt(1.0 + (1.0 / eps)) * cbrt(1.0 + (1.0 / eps))) * (cbrt(1.0 + (1.0 / eps)) / exp(x * (1.0 - eps)))) - (((1.0 / eps) - 1.0) * exp(x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 244.84131261463813

   1. Initial program 39.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666666 \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]

   3. Simplified 1.3

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x}}{2}\]
   4. Using strategy rm

   5. Applied associate--l+_binary64_15 1.3

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.6666666666666666 + \left(2 - x \cdot x\right)}}{2}\]
   6. Using strategy rm

   7. Applied flip3-+_binary64_81 1.3

      \[\leadsto \frac{\color{blue}{\frac{{\left({x}^{3} \cdot 0.6666666666666666\right)}^{3} + {\left(2 - x \cdot x\right)}^{3}}{\left({x}^{3} \cdot 0.6666666666666666\right) \cdot \left({x}^{3} \cdot 0.6666666666666666\right) + \left(\left(2 - x \cdot x\right) \cdot \left(2 - x \cdot x\right) - \left({x}^{3} \cdot 0.6666666666666666\right) \cdot \left(2 - x \cdot x\right)\right)}}}{2}\]

   8. Simplified 1.3

      \[\leadsto \frac{\frac{\color{blue}{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\left({x}^{3} \cdot 0.6666666666666666\right) \cdot \left({x}^{3} \cdot 0.6666666666666666\right) + \left(\left(2 - x \cdot x\right) \cdot \left(2 - x \cdot x\right) - \left({x}^{3} \cdot 0.6666666666666666\right) \cdot \left(2 - x \cdot x\right)\right)}}{2}\]

   9. Simplified 1.3

      \[\leadsto \frac{\frac{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}{\color{blue}{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}{2}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt_binary64_100 1.3

       \[\leadsto \frac{\frac{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}{\color{blue}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)} \cdot \sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}}{2}\]

   12. Applied add-cube-cbrt_binary64_113 1.3

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}} \cdot \sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}\right) \cdot \sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)} \cdot \sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}{2}\]

   13. Applied times-frac_binary64_84 1.3

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}} \cdot \sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}} \cdot \frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}}{2}\]

   if 244.84131261463813 < x

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt_binary64_113 0.1

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   4. Applied associate-*l*_binary64_19 0.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   5. Simplified 0.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 244.84131261463813:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}} \cdot \sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}} \cdot \frac{\sqrt[3]{{x}^{9} \cdot 0.2962962962962963 + {\left(2 - x \cdot x\right)}^{3}}}{\sqrt{{x}^{6} \cdot 0.4444444444444444 + \left(2 - x \cdot x\right) \cdot \left(\left(2 - x \cdot x\right) - {x}^{3} \cdot 0.6666666666666666\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \frac{\sqrt[3]{1 + \frac{1}{\varepsilon}}}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))