expfmod

Average Error: 59.6 → 23.9

Time: 9.7s

Precision: binary64

Math

\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 1\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt[3]{{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}}}\\ \end{array}\]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (if (or (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.0)
         (not (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 1.0)))
   (exp (- x))
   (exp (cbrt (pow (log (/ (fmod (exp x) (sqrt (cos x))) (exp x))) 3.0)))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	double tmp;
	if (((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0) || !((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 1.0)) {
		tmp = exp(-x);
	} else {
		tmp = exp(cbrt(pow(log(fmod(exp(x), sqrt(cos(x))) / exp(x)), 3.0)));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0 or 1 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 61.8

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]

   2. Simplified 61.8

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\]
   3. Using strategy rm

   4. Applied add-exp-log_binary64_798 61.8

      \[\leadsto \color{blue}{e^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}}\]

   5. Simplified 61.8

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}\]

   6. Taylor expanded around inf 24.4

      \[\leadsto e^{\color{blue}{-1 \cdot x}}\]

   7. Simplified 24.4

      \[\leadsto e^{\color{blue}{-x}}\]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1

   1. Initial program 12.8

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]

   2. Simplified 12.7

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\]
   3. Using strategy rm

   4. Applied add-exp-log_binary64_798 12.7

      \[\leadsto \color{blue}{e^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}}\]

   5. Simplified 12.7

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube_binary64_796 12.7

      \[\leadsto e^{\color{blue}{\sqrt[3]{\left(\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right) \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right) \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}}}\]

   8. Simplified 12.7

      \[\leadsto e^{\sqrt[3]{\color{blue}{{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}^{3}}}}\]
   9. Using strategy rm

   10. Applied add-log-exp_binary64_799 12.7

       \[\leadsto e^{\sqrt[3]{{\color{blue}{\log \left(e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}}^{3}}}\]

   11. Simplified 12.7

       \[\leadsto e^{\sqrt[3]{{\log \color{blue}{\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}}^{3}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 1\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt[3]{{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))