expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 99.1%

Time: 12.9s

Alternatives: 6

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.9628786964785472e+264

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: 6.8% 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: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e-309)
   (fmod
    1.0
    (fma
     x
     (* x (sqrt 0.041666666666666664))
     (/ -0.25 (sqrt 0.041666666666666664))))
   (if (<= x 0.5)
     (*
      (fmod
       (fma (* x x) (* x (+ 0.16666666666666666 (/ 0.5 x))) x)
       (sqrt (fma (* x x) (fma x (* x 0.041666666666666664) -0.5) 1.0)))
      (- 1.0 x))
     (* (fmod 1.0 (sqrt (cos x))) (exp (- x))))))

C

double code(double x) {
	double tmp;
	if (x <= 3e-309) {
		tmp = fmod(1.0, fma(x, (x * sqrt(0.041666666666666664)), (-0.25 / sqrt(0.041666666666666664))));
	} else if (x <= 0.5) {
		tmp = fmod(fma((x * x), (x * (0.16666666666666666 + (0.5 / x))), x), sqrt(fma((x * x), fma(x, (x * 0.041666666666666664), -0.5), 1.0))) * (1.0 - x);
	} else {
		tmp = fmod(1.0, sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e-309)
		tmp = rem(1.0, fma(x, Float64(x * sqrt(0.041666666666666664)), Float64(-0.25 / sqrt(0.041666666666666664))));
	elseif (x <= 0.5)
		tmp = Float64(rem(fma(Float64(x * x), Float64(x * Float64(0.16666666666666666 + Float64(0.5 / x))), x), sqrt(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -0.5), 1.0))) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3e-309], N[With[{TMP1 = 1.0, TMP2 = N[(x * N[(x * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.000000000000001e-309

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 9.7

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 9.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + \sqrt{\frac{1}{24}}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + {x}^{2} \cdot \sqrt{\frac{1}{24}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 98.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(1, \frac{-0.25}{\sqrt{0.041666666666666664}}, x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{x \cdot \left(x \cdot \sqrt{\frac{1}{24}}\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(x \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}} \cdot x}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{\sqrt{\frac{1}{24}}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\color{blue}{\frac{-1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       15. lower-sqrt.f64 95.3

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\color{blue}{\sqrt{0.041666666666666664}}}\right)\right)\right) \]

   11. Simplified 95.3%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right) \]

   if 3.000000000000001e-309 < x < 0.5

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 6.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 6.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

      3. lower--.f64 6.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   8. Simplified 6.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. lower-fma.f64 6.3

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   11. Simplified 6.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   13. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \frac{1}{{x}^{2}}\right)}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{1}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       10. cube-mult N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       11. unpow3 N/A

           \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       14. *-rgt-identity N/A

           \[\leadsto \left(\left({x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + \color{blue}{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   14. Simplified 98.1%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   if 0.5 < x

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

   5. Simplified 98.6%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e-309)
   (fmod
    1.0
    (fma
     x
     (* x (sqrt 0.041666666666666664))
     (/ -0.25 (sqrt 0.041666666666666664))))
   (if (<= x 0.5)
     (*
      (fmod
       (fma (* x x) (* x (+ 0.16666666666666666 (/ 0.5 x))) x)
       (sqrt (fma (* x x) (fma x (* x 0.041666666666666664) -0.5) 1.0)))
      (- 1.0 x))
     (* (exp (- x)) (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 3e-309) {
		tmp = fmod(1.0, fma(x, (x * sqrt(0.041666666666666664)), (-0.25 / sqrt(0.041666666666666664))));
	} else if (x <= 0.5) {
		tmp = fmod(fma((x * x), (x * (0.16666666666666666 + (0.5 / x))), x), sqrt(fma((x * x), fma(x, (x * 0.041666666666666664), -0.5), 1.0))) * (1.0 - x);
	} else {
		tmp = exp(-x) * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e-309)
		tmp = rem(1.0, fma(x, Float64(x * sqrt(0.041666666666666664)), Float64(-0.25 / sqrt(0.041666666666666664))));
	elseif (x <= 0.5)
		tmp = Float64(rem(fma(Float64(x * x), Float64(x * Float64(0.16666666666666666 + Float64(0.5 / x))), x), sqrt(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -0.5), 1.0))) * Float64(1.0 - x));
	else
		tmp = Float64(exp(Float64(-x)) * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3e-309], N[With[{TMP1 = 1.0, TMP2 = N[(x * N[(x * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.000000000000001e-309

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 9.7

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 9.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + \sqrt{\frac{1}{24}}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + {x}^{2} \cdot \sqrt{\frac{1}{24}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 98.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(1, \frac{-0.25}{\sqrt{0.041666666666666664}}, x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{x \cdot \left(x \cdot \sqrt{\frac{1}{24}}\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(x \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}} \cdot x}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{\sqrt{\frac{1}{24}}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\color{blue}{\frac{-1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       15. lower-sqrt.f64 95.3

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\color{blue}{\sqrt{0.041666666666666664}}}\right)\right)\right) \]

   11. Simplified 95.3%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right) \]

   if 3.000000000000001e-309 < x < 0.5

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 6.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 6.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

      3. lower--.f64 6.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   8. Simplified 6.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. lower-fma.f64 6.3

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   11. Simplified 6.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   13. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \frac{1}{{x}^{2}}\right)}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{1}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       10. cube-mult N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       11. unpow3 N/A

           \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       14. *-rgt-identity N/A

           \[\leadsto \left(\left({x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + \color{blue}{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   14. Simplified 98.1%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   if 0.5 < x

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 98.4

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

   8. Simplified 98.4%

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \cdot e^{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e-309)
   (fmod
    1.0
    (fma
     x
     (* x (sqrt 0.041666666666666664))
     (/ -0.25 (sqrt 0.041666666666666664))))
   (if (<= x 0.5)
     (*
      (fmod
       (fma (* x x) (* x (+ 0.16666666666666666 (/ 0.5 x))) x)
       (sqrt (fma (* x x) (fma x (* x 0.041666666666666664) -0.5) 1.0)))
      (- 1.0 x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 3e-309) {
		tmp = fmod(1.0, fma(x, (x * sqrt(0.041666666666666664)), (-0.25 / sqrt(0.041666666666666664))));
	} else if (x <= 0.5) {
		tmp = fmod(fma((x * x), (x * (0.16666666666666666 + (0.5 / x))), x), sqrt(fma((x * x), fma(x, (x * 0.041666666666666664), -0.5), 1.0))) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e-309)
		tmp = rem(1.0, fma(x, Float64(x * sqrt(0.041666666666666664)), Float64(-0.25 / sqrt(0.041666666666666664))));
	elseif (x <= 0.5)
		tmp = Float64(rem(fma(Float64(x * x), Float64(x * Float64(0.16666666666666666 + Float64(0.5 / x))), x), sqrt(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -0.5), 1.0))) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3e-309], N[With[{TMP1 = 1.0, TMP2 = N[(x * N[(x * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.000000000000001e-309

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 9.7

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 9.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + \sqrt{\frac{1}{24}}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + {x}^{2} \cdot \sqrt{\frac{1}{24}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 98.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(1, \frac{-0.25}{\sqrt{0.041666666666666664}}, x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{x \cdot \left(x \cdot \sqrt{\frac{1}{24}}\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(x \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}} \cdot x}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{\sqrt{\frac{1}{24}}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\color{blue}{\frac{-1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       15. lower-sqrt.f64 95.3

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\color{blue}{\sqrt{0.041666666666666664}}}\right)\right)\right) \]

   11. Simplified 95.3%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right) \]

   if 3.000000000000001e-309 < x < 0.5

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 6.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 6.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

      3. lower--.f64 6.3

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   8. Simplified 6.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. lower-fma.f64 6.3

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   11. Simplified 6.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]
   13. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \frac{1}{{x}^{2}}\right)}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{1}\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot 1\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       10. cube-mult N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       11. unpow3 N/A

           \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(\left(\color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)} + x \cdot 1\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       14. *-rgt-identity N/A

           \[\leadsto \left(\left({x}^{2} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)\right) + \color{blue}{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   14. Simplified 98.1%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right)} \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right) \]

   if 0.5 < x

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right)} \]
   7. Step-by-step derivation

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.1

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
   10. Step-by-step derivation

   11. Simplified 98.2%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, x \cdot \left(0.16666666666666666 + \frac{0.5}{x}\right), x\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 400:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 400.0)
   (fmod
    1.0
    (fma
     x
     (* x (sqrt 0.041666666666666664))
     (/ -0.25 (sqrt 0.041666666666666664))))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 400.0) {
		tmp = fmod(1.0, fma(x, (x * sqrt(0.041666666666666664)), (-0.25 / sqrt(0.041666666666666664))));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 400.0)
		tmp = rem(1.0, fma(x, Float64(x * sqrt(0.041666666666666664)), Float64(-0.25 / sqrt(0.041666666666666664))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 400.0], N[With[{TMP1 = 1.0, TMP2 = N[(x * N[(x * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 400

   1. Initial program 8.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 8.3

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Simplified 8.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + \sqrt{\frac{1}{24}}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right) + {x}^{2} \cdot \sqrt{\frac{1}{24}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 51.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(1, \frac{-0.25}{\sqrt{0.041666666666666664}}, x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left({x}^{2} \cdot \sqrt{\frac{1}{24}} - \frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt{\frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{x \cdot \left(x \cdot \sqrt{\frac{1}{24}}\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(x \cdot \color{blue}{\left(\sqrt{\frac{1}{24}} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}} \cdot x}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \color{blue}{\sqrt{\frac{1}{24}}} \cdot x, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{\sqrt{\frac{1}{24}}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right)\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\color{blue}{\frac{-1}{4}}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \color{blue}{\frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}}\right)\right)\right) \]

       15. lower-sqrt.f64 50.5

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\color{blue}{\sqrt{0.041666666666666664}}}\right)\right)\right) \]

   11. Simplified 50.5%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{\frac{1}{24}} \cdot x, \frac{\frac{-1}{4}}{\sqrt{\frac{1}{24}}}\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 52.8%

       \[\leadsto \left(\color{blue}{1} \bmod \left(\mathsf{fma}\left(x, \sqrt{0.041666666666666664} \cdot x, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right) \]

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right)} \]
   7. Step-by-step derivation

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
   10. Step-by-step derivation

   11. Simplified 100.0%

       \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 400:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x, x \cdot \sqrt{0.041666666666666664}, \frac{-0.25}{\sqrt{0.041666666666666664}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 24.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + 1\right) \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod (+ x 1.0) 1.0))

C

double code(double x) {
	return fmod((x + 1.0), 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod((x + 1.0d0), 1.0d0)
end function

Python

def code(x):
	return math.fmod((x + 1.0), 1.0)

Julia

function code(x)
	return rem(Float64(x + 1.0), 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
4. Step-by-step derivation

5. Simplified 6.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right)} \]
7. Step-by-step derivation

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 5.2

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

8. Simplified 5.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod 1\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod 1\right) \]

    2. lower-+.f64 25.1

       \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod 1\right) \]

11. Simplified 25.1%

    \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod 1\right) \]
12. Add Preprocessing

Alternative 6: 23.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod 1.0 1.0))

C

double code(double x) {
	return fmod(1.0, 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(1.0d0, 1.0d0)
end function

Python

def code(x):
	return math.fmod(1.0, 1.0)

Julia

function code(x)
	return rem(1.0, 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 7.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
4. Step-by-step derivation

5. Simplified 6.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right)} \]
7. Step-by-step derivation

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 5.2

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

8. Simplified 5.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
10. Step-by-step derivation

11. Simplified 23.6%

    \[\leadsto \left(\color{blue}{1} \bmod 1\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))