expfmod (used to be hard to sample)

Percentage Accurate: 6.4% → 63.1%

Time: 13.5s

Alternatives: 13

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.4206958696063833e+300

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: 6.4% 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: 63.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0026041666666666665, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (fma
          (* x x)
          (fma
           (* x x)
           (fma x (* x -0.0026041666666666665) -0.010416666666666666)
           -0.25)
          1.0)))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma((x * x), fma(x, (x * -0.0026041666666666665), -0.010416666666666666), -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.0026041666666666665), -0.010416666666666666), -0.25), 1.0)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0026041666666666665), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}, \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{384} \cdot {x}^{2} - \frac{1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{384} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{384}} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{384} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{384}\right)} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{-1}{384}\right) + \color{blue}{\frac{-1}{96}}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{384}, \frac{-1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      16. lower-*.f64 85.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.0026041666666666665}, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   8. Simplified 85.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0026041666666666665, -0.010416666666666666\right), -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0026041666666666665, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (sqrt (fma (* x x) (fma (* x x) 0.041666666666666664 -0.5) 1.0))))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), sqrt(fma((x * x), fma((x * x), 0.041666666666666664, -0.5), 1.0)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), sqrt(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Simplified 84.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (fma (* x x) (fma x (* x -0.010416666666666666) -0.25) 1.0)))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma(x, (x * -0.010416666666666666), -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * -0.010416666666666666), -0.25), 1.0)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{96} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{96}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{96} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{96}\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{-1}{96}\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 84.3

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.010416666666666666}, -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   8. Simplified 84.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (<= t_1 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_1 2.0)
       (* t_0 (fmod (exp x) (sqrt (fma x (* x -0.5) 1.0))))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), sqrt(fma(x, (x * -0.5), 1.0)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), sqrt(fma(x, Float64(x * -0.5), 1.0))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.6

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

   5. Simplified 80.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fmod (exp x) (sqrt (cos x))) (exp (- x)))))
   (if (<= t_0 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_0 2.0)
       (*
        (fmod
         (exp x)
         (sqrt (fma (* x x) (fma (* x x) 0.041666666666666664 -0.5) 1.0)))
        (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_0 <= 2.0) {
		tmp = fmod(exp(x), sqrt(fma((x * x), fma((x * x), 0.041666666666666664, -0.5), 1.0))) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_0 <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0))) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 75.1

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

   8. Simplified 75.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fmod (exp x) (sqrt (cos x))) (exp (- x)))))
   (if (<= t_0 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_0 2.0)
       (*
        (fmod (exp x) (sqrt (fma x (* x -0.5) 1.0)))
        (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_0 <= 2.0) {
		tmp = fmod(exp(x), sqrt(fma(x, (x * -0.5), 1.0))) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_0 <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(fma(x, Float64(x * -0.5), 1.0))) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 73.7

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

   8. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fmod (exp x) (sqrt (cos x))) (exp (- x)))))
   (if (<= t_0 0.0)
     (fmod
      (exp x)
      (/ -1.5 (* (sqrt 0.041666666666666664) (* x (* x (* x x))))))
     (if (<= t_0 2.0)
       (*
        (fmod (fma x (fma x 0.5 1.0) 1.0) (sqrt (fma x (* x -0.5) 1.0)))
        (fma x (fma x 0.5 -1.0) 1.0))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fmod(exp(x), (-1.5 / (sqrt(0.041666666666666664) * (x * (x * (x * x))))));
	} else if (t_0 <= 2.0) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), sqrt(fma(x, (x * -0.5), 1.0))) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = rem(exp(x), Float64(-1.5 / Float64(sqrt(0.041666666666666664) * Float64(x * Float64(x * Float64(x * x))))));
	elseif (t_0 <= 2.0)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), sqrt(fma(x, Float64(x * -0.5), 1.0))) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[Sqrt[0.041666666666666664], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.1%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.1

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

   5. Simplified 4.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-fmod.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right)} \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{{x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{\left(x \cdot \left(x \cdot {x}^{2}\right)\right)} \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \sqrt{\frac{1}{24}}}\right)\right) \]

       13. lower-sqrt.f64 55.4

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \]

   14. Simplified 55.4%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 68.0

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

   8. Simplified 68.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 69.4

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

   11. Simplified 69.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\sqrt{0.041666666666666664} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      t_0
      (fmod
       (exp x)
       (/ -1.5 (* (* (* x x) (* x x)) (sqrt 0.041666666666666664)))))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = t_0 * fmod(exp(x), (-1.5 / (((x * x) * (x * x)) * sqrt(0.041666666666666664))));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = t_0 * mod(exp(x), ((-1.5d0) / (((x * x) * (x * x)) * sqrt(0.041666666666666664d0))))
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = t_0 * math.fmod(math.exp(x), (-1.5 / (((x * x) * (x * x)) * math.sqrt(0.041666666666666664))))
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), Float64(-1.5 / Float64(Float64(Float64(x * x) * Float64(x * x)) * sqrt(0.041666666666666664)))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-1.5 / N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.1

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

   5. Simplified 10.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left(\left(\sqrt{\frac{1}{24}} + \left(\frac{1}{8} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{6} \cdot {\left(\sqrt{\frac{1}{24}}\right)}^{3}} + \frac{1}{2} \cdot \frac{1 - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{24}}\right)}^{2}}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)\right) - \frac{\frac{1}{4}}{{x}^{2} \cdot \sqrt{\frac{1}{24}}}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   8. Simplified 55.7%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\sqrt{0.041666666666666664} + \left(\frac{-0.25}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{0.041666666666666664}} + \frac{-1.5}{{x}^{6} \cdot \sqrt{0.041666666666666664}}\right)\right) + \frac{-0.25}{x \cdot \left(x \cdot \sqrt{0.041666666666666664}\right)}\right)\right)\right)}\right) \cdot e^{-x} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{\frac{-3}{2}}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{\frac{-3}{2}}{\color{blue}{{x}^{4} \cdot \sqrt{\frac{1}{24}}}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sqrt.f64 55.7

           \[\leadsto \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\sqrt{0.041666666666666664}}}\right)\right) \cdot e^{-x} \]

   11. Simplified 55.7%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\frac{-1.5}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{0.041666666666666664}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)}\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (*
    (fmod (fma x (fma x 0.5 1.0) 1.0) (sqrt (fma x (* x -0.5) 1.0)))
    (fma x (fma x 0.5 -1.0) 1.0))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), sqrt(fma(x, (x * -0.5), 1.0))) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), sqrt(fma(x, Float64(x * -0.5), 1.0))) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 2.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 9.8

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

   5. Simplified 9.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 8.9

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

   8. Simplified 8.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 9.0

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

   11. Simplified 9.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

Alternative 10: 26.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(x + 1\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (* (- 1.0 x) (fmod (+ x 1.0) (fma x (* x -0.25) 1.0)))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = (1.0 - x) * fmod((x + 1.0), fma(x, (x * -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(Float64(1.0 - x) * rem(Float64(x + 1.0), fma(x, Float64(x * -0.25), 1.0)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 2.0], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fmod.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 8.2

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 8.2

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

   8. Simplified 8.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 7.8

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

   11. Simplified 7.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 96.4%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(x + 1\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.01)
   (* (fmod (fma x (fma x 0.5 1.0) 1.0) (fma x (* x -0.25) 1.0)) (- 1.0 x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.01) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), fma(x, (x * -0.25), 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.01)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), fma(x, Float64(x * -0.25), 1.0)) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.01], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $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 2 regimes

2. if x < 0.0100000000000000002

   1. Initial program 10.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fmod.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 8.1

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

   5. Simplified 8.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 8.1

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

   8. Simplified 8.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

       5. lower-fma.f64 8.2

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \left(1 - x\right) \]

   11. Simplified 8.2%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \left(1 - x\right) \]

   if 0.0100000000000000002 < 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

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 0.0%

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

   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. Add Preprocessing

Alternative 12: 25.5% 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 8.0%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.6

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

5. Simplified 5.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 5.6%

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

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 24.8

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

11. Simplified 24.8%

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

Alternative 13: 24.5% 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 8.0%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.6

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

5. Simplified 5.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 5.6%

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

9. Taylor expanded in x around 0

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

11. Simplified 23.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))