expfmod

Average Error: 59.6 → 23.5

Time: 14.3s

Precision: binary64

Cost: 77708

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{{t_0}^{2}}{\frac{e^{x}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (fmod (exp x) (sqrt (cos x))))) (t_1 (exp (- x))))
   (if (<= x -2e-16)
     (/ (fmod (exp x) 1.0) (exp x))
     (if (<= x -5e-310)
       t_1
       (if (<= x 0.05096768893116153)
         (/ (pow t_0 2.0) (/ (exp x) t_0))
         t_1)))))

C

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

↓

double code(double x) {
	double t_0 = cbrt(fmod(exp(x), sqrt(cos(x))));
	double t_1 = exp(-x);
	double tmp;
	if (x <= -2e-16) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else if (x <= -5e-310) {
		tmp = t_1;
	} else if (x <= 0.05096768893116153) {
		tmp = pow(t_0, 2.0) / (exp(x) / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = cbrt(rem(exp(x), sqrt(cos(x))))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-16)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	elseif (x <= -5e-310)
		tmp = t_1;
	elseif (x <= 0.05096768893116153)
		tmp = Float64((t_0 ^ 2.0) / Float64(exp(x) / t_0));
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = N[Power[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-310], t$95$1, If[LessEqual[x, 0.05096768893116153], N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-16

   1. Initial program 8.7

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

   2. Simplified 8.5

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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)))): 1 points increase in error, 2 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. Taylor expanded in x around 0 8.8

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

   if -2e-16 < x < -4.999999999999985e-310 or 0.0509676889311615269 < x

   1. Initial program 62.3

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

   2. Simplified 62.3

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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)))): 1 points increase in error, 2 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 62.3

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

   4. Taylor expanded in x around inf 0.3

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

   5. Simplified 0.3

      \[\leadsto e^{\color{blue}{-x}} \]
      Proof
      (neg.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)): 0 points increase in error, 0 points decrease in error

   if -4.999999999999985e-310 < x < 0.0509676889311615269

   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)))): 1 points increase in error, 2 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

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

   4. Applied egg-rr 58.9

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

4. Final simplification 23.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}}{\frac{e^{x}}{\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.5
Cost	77708

\[\begin{array}{l} t_0 := \sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{t_0 \cdot {t_0}^{2}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.5
Cost	45516

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{3}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	23.5
Cost	32908

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	23.5
Cost	32844

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \frac{-1}{-e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	23.5
Cost	32780

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.5
Cost	32652

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05096768893116153:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	24.2
Cost	19588

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 8
Error	24.8
Cost	6528

\[e^{-x} \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))