expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 62.6%

Time: 43.9s

Alternatives: 2

Speedup: 5.0×

• could not determine a ground truth

  1. x = -3.392392945647322e+114

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Alternative 1: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(t\_0 \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e^{t\_0}}\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) t_0) t_1) 2.0)
     (/
      (fmod (exp x) (+ (* t_0 0.6666666666666666) (log (cbrt (exp t_0)))))
      (exp x))
     t_1)))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), t_0) * t_1) <= 2.0) {
		tmp = fmod(exp(x), ((t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), t_0) * t_1) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(Float64(t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(t$95$0 * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], t$95$1]]]

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))) < 2

   1. Initial program 8.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. /-rgt-identity 8.1%

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

      2. associate-/r/ 8.1%

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

      3. exp-neg 8.1%

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

      4. remove-double-neg 8.1%

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

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 8.1%

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

      2. add-cube-cbrt 55.8%

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

      3. log-prod 55.7%

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

      4. pow2 55.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      5. pow1/2 55.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{\color{blue}{{\cos x}^{0.5}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      6. pow1/2 55.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{{\cos x}^{0.5}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{{\cos x}^{0.5}}}}\right)\right)\right)}{e^{x}} \]

   6. Applied egg-rr 55.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{{\cos x}^{0.5}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{{\cos x}^{0.5}}}\right)\right)}\right)}{e^{x}} \]
   7. Step-by-step derivation

      1. unpow1/2 55.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{\color{blue}{\sqrt{\cos x}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{{\cos x}^{0.5}}}\right)\right)\right)}{e^{x}} \]

      2. unpow1/2 55.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\sqrt{\cos x}}}}\right)\right)\right)}{e^{x}} \]

   8. Simplified 55.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]
   9. Step-by-step derivation

      1. unpow2 55.7%

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

      2. log-prod 55.8%

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

      3. pow1/3 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \color{blue}{\left({\left(e^{\sqrt{\cos x}}\right)}^{0.3333333333333333}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      4. pow1/2 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left({\left(e^{\color{blue}{{\cos x}^{0.5}}}\right)}^{0.3333333333333333}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      5. metadata-eval 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left({\left(e^{{\cos x}^{\color{blue}{\left(0.16666666666666666 \cdot 3\right)}}}\right)}^{0.3333333333333333}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      6. pow-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left({\left(e^{\color{blue}{{\left({\cos x}^{0.16666666666666666}\right)}^{3}}}\right)}^{0.3333333333333333}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      7. log-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\color{blue}{0.3333333333333333 \cdot \log \left(e^{{\left({\cos x}^{0.16666666666666666}\right)}^{3}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      8. add-log-exp 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot \color{blue}{{\left({\cos x}^{0.16666666666666666}\right)}^{3}} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      9. pow-pow 55.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot \color{blue}{{\cos x}^{\left(0.16666666666666666 \cdot 3\right)}} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      10. metadata-eval 55.8%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{\color{blue}{0.5}} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      11. pow1/3 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{0.5} + \log \color{blue}{\left({\left(e^{\sqrt{\cos x}}\right)}^{0.3333333333333333}\right)}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      12. pow1/2 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{0.5} + \log \left({\left(e^{\color{blue}{{\cos x}^{0.5}}}\right)}^{0.3333333333333333}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      13. metadata-eval 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{0.5} + \log \left({\left(e^{{\cos x}^{\color{blue}{\left(0.16666666666666666 \cdot 3\right)}}}\right)}^{0.3333333333333333}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      14. pow-pow 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{0.5} + \log \left({\left(e^{\color{blue}{{\left({\cos x}^{0.16666666666666666}\right)}^{3}}}\right)}^{0.3333333333333333}\right)\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

      15. log-pow 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(0.3333333333333333 \cdot {\cos x}^{0.5} + \color{blue}{0.3333333333333333 \cdot \log \left(e^{{\left({\cos x}^{0.16666666666666666}\right)}^{3}}\right)}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. Applied egg-rr 55.9%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(0.3333333333333333 \cdot {\cos x}^{0.5} + 0.3333333333333333 \cdot {\cos x}^{0.5}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
   11. Step-by-step derivation

       1. distribute-rgt-out 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\cos x}^{0.5} \cdot \left(0.3333333333333333 + 0.3333333333333333\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

       2. unpow1/2 55.9%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x}} \cdot \left(0.3333333333333333 + 0.3333333333333333\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

       3. metadata-eval 55.9%

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

   12. Simplified 55.9%

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

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

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
   6. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(-0.25 \cdot {x}^{2} + 1\right)}\right)}{e^{x}} \]

      2. fma-define 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.25, {x}^{2}, 1\right)\right)}\right)}{e^{x}} \]

      3. unpow2 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, \color{blue}{x \cdot x}, 1\right)\right)\right)}{e^{x}} \]

   7. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)}\right)}{e^{x}} \]
   8. Step-by-step derivation

      1. add-exp-log 0.0%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\right)}} \]

      2. log-div 0.0%

         \[\leadsto e^{\color{blue}{\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) - \log \left(e^{x}\right)}} \]

      3. fma-undefine 0.0%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left(-0.25 \cdot \left(x \cdot x\right) + 1\right)}\right) - \log \left(e^{x}\right)} \]

      4. +-commutative 0.0%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot \left(x \cdot x\right)\right)}\right) - \log \left(e^{x}\right)} \]

      5. associate-*r* 0.0%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\left(-0.25 \cdot x\right) \cdot x}\right)\right) - \log \left(e^{x}\right)} \]

      6. add-log-exp 0.0%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \left(-0.25 \cdot x\right) \cdot x\right)\right) - \color{blue}{x}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \left(-0.25 \cdot x\right) \cdot x\right)\right) - x}} \]

   10. Taylor expanded in x around inf 100.0%

       \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   11. Step-by-step derivation

       1. mul-1-neg 100.0%

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

   12. Simplified 100.0%

       \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 60.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

code[x_] := N[Exp[(-x)], $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Step-by-step derivation

   1. /-rgt-identity 6.4%

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

   2. associate-/r/ 6.4%

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

   3. exp-neg 6.4%

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

   4. remove-double-neg 6.4%

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

3. Simplified 6.4%

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

5. Taylor expanded in x around 0 5.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
6. Step-by-step derivation

   1. +-commutative 5.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(-0.25 \cdot {x}^{2} + 1\right)}\right)}{e^{x}} \]

   2. fma-define 5.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.25, {x}^{2}, 1\right)\right)}\right)}{e^{x}} \]

   3. unpow2 5.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, \color{blue}{x \cdot x}, 1\right)\right)\right)}{e^{x}} \]

7. Simplified 5.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)}\right)}{e^{x}} \]
8. Step-by-step derivation

   1. add-exp-log 5.8%

      \[\leadsto \color{blue}{e^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right)}{e^{x}}\right)}} \]

   2. log-div 5.8%

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) - \log \left(e^{x}\right)}} \]

   3. fma-undefine 5.8%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left(-0.25 \cdot \left(x \cdot x\right) + 1\right)}\right) - \log \left(e^{x}\right)} \]

   4. +-commutative 5.8%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot \left(x \cdot x\right)\right)}\right) - \log \left(e^{x}\right)} \]

   5. associate-*r* 5.8%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\left(-0.25 \cdot x\right) \cdot x}\right)\right) - \log \left(e^{x}\right)} \]

   6. add-log-exp 5.8%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \left(-0.25 \cdot x\right) \cdot x\right)\right) - \color{blue}{x}} \]

9. Applied egg-rr 5.8%

   \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(1 + \left(-0.25 \cdot x\right) \cdot x\right)\right) - x}} \]

10. Taylor expanded in x around inf 63.2%

    \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
11. Step-by-step derivation

    1. mul-1-neg 63.2%

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

12. Simplified 63.2%

    \[\leadsto e^{\color{blue}{-x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))