expfmod

Average Error: 59.5 → 23.7

Time: 16.4s

Precision: binary64

Cost: 32776

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.820084872909202 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5186950637553355:\\ \;\;\;\;\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4.820084872909202e-299)
   1.0
   (if (<= x 0.5186950637553355)
     (+ (+ 1.0 (/ (fmod (exp x) (sqrt (cos x))) (exp x))) -1.0)
     (exp (- x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -4.820084872909202e-299) {
		tmp = 1.0;
	} else if (x <= 0.5186950637553355) {
		tmp = (1.0 + (fmod(exp(x), sqrt(cos(x))) / exp(x))) + -1.0;
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.820084872909202d-299)) then
        tmp = 1.0d0
    else if (x <= 0.5186950637553355d0) then
        tmp = (1.0d0 + (mod(exp(x), sqrt(cos(x))) / exp(x))) + (-1.0d0)
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

↓

def code(x):
	tmp = 0
	if x <= -4.820084872909202e-299:
		tmp = 1.0
	elif x <= 0.5186950637553355:
		tmp = (1.0 + (math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x))) + -1.0
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -4.820084872909202e-299)
		tmp = 1.0;
	elseif (x <= 0.5186950637553355)
		tmp = Float64(Float64(1.0 + Float64(rem(exp(x), sqrt(cos(x))) / exp(x))) + -1.0);
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -4.820084872909202e-299], 1.0, If[LessEqual[x, 0.5186950637553355], N[(N[(1.0 + N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8200848729092019e-299

   1. Initial program 58.2

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

   2. Simplified 58.2

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      Proof
      (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) 1)) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (/.f64 1 (exp.f64 x)))): 3 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.1

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

   4. Applied egg-rr 58.2

      \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}\right)} - x} \]

   5. Applied egg-rr 0

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

   6. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{1} \]

   if -4.8200848729092019e-299 < x < 0.518695063755335473

   1. Initial program 58.9

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

   2. Simplified 58.9

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      Proof
      (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) 1)) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (/.f64 1 (exp.f64 x)))): 3 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.9

      \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) - 1} \]

   if 0.518695063755335473 < x

   1. Initial program 63.5

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

   2. Simplified 63.5

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      Proof
      (/.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) 1)) (exp.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (/.f64 1 (exp.f64 x)))): 3 points increase in error, 0 points decrease in error
      (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 x)))): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 63.5

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

   4. Taylor expanded in x around inf 0.7

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

   5. Simplified 0.7

      \[\leadsto e^{\color{blue}{-x}} \]
      Proof
      (neg.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.820084872909202 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5186950637553355:\\ \;\;\;\;\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.1
Cost	90564

\[\begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq 0.5186950637553355:\\ \;\;\;\;{\left({e}^{\left({t_0}^{2} - x \cdot x\right)}\right)}^{\left(\frac{1}{x + t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2
Error	23.7
Cost	32520

\[\begin{array}{l} \mathbf{if}\;x \leq -4.820084872909202 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5186950637553355:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 3
Error	23.6
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3579569753336683 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 4
Error	36.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))