expfmod

Average Error: 59.6 → 37.1

Time: 16.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\cos x}\\ \frac{\left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{0.6666666666666666 \cdot t_0}\right)}^{3} + \log \left(\sqrt[3]{e^{t_0}}\right)\right)\right)}{e^{x}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (/
    (fmod
     (exp x)
     (+ (pow (cbrt (* 0.6666666666666666 t_0)) 3.0) (log (cbrt (exp t_0)))))
    (exp x))))

C

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

↓

double code(double x) {
	double t_0 = sqrt(cos(x));
	return fmod(exp(x), (pow(cbrt((0.6666666666666666 * t_0)), 3.0) + log(cbrt(exp(t_0))))) / exp(x);
}

Error

Bits error versus x

Derivation

1. Initial program 59.6

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

2. Simplified 59.6

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

3. Applied egg-rr 37.2

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

4. Applied egg-rr 37.1

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot \sqrt{\cos x}}\right)}^{3}} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

5. Final simplification 37.1

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{0.6666666666666666 \cdot \sqrt{\cos x}}\right)}^{3} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))