expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 61.8%

Time: 13.7s

Alternatives: 9

Speedup: 4.1×

• could not determine a ground truth

  1. x = -2.9769563468390585e+128

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: 6.6% 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: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(\sqrt{0.041666666666666664} + \frac{-0.25}{\left(x \cdot x\right) \cdot \sqrt{0.041666666666666664}}\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1e-154)
     (*
      (fmod
       (exp x)
       (*
        (* x x)
        (+
         (sqrt 0.041666666666666664)
         (/ -0.25 (* (* x x) (sqrt 0.041666666666666664))))))
      t_0)
     (if (<= x -2e-310)
       (*
        t_0
        (fmod
         (exp x)
         (* x (* x (* x (* x (+ -0.010416666666666666 (/ -0.25 (* x x)))))))))
       (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1e-154) {
		tmp = fmod(exp(x), ((x * x) * (sqrt(0.041666666666666664) + (-0.25 / ((x * x) * sqrt(0.041666666666666664)))))) * t_0;
	} else if (x <= -2e-310) {
		tmp = t_0 * fmod(exp(x), (x * (x * (x * (x * (-0.010416666666666666 + (-0.25 / (x * x))))))));
	} else {
		tmp = fmod((x + 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 (x <= (-1d-154)) then
        tmp = mod(exp(x), ((x * x) * (sqrt(0.041666666666666664d0) + ((-0.25d0) / ((x * x) * sqrt(0.041666666666666664d0)))))) * t_0
    else if (x <= (-2d-310)) then
        tmp = t_0 * mod(exp(x), (x * (x * (x * (x * ((-0.010416666666666666d0) + ((-0.25d0) / (x * x))))))))
    else
        tmp = mod((x + 1.0d0), 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -1e-154:
		tmp = math.fmod(math.exp(x), ((x * x) * (math.sqrt(0.041666666666666664) + (-0.25 / ((x * x) * math.sqrt(0.041666666666666664)))))) * t_0
	elif x <= -2e-310:
		tmp = t_0 * math.fmod(math.exp(x), (x * (x * (x * (x * (-0.010416666666666666 + (-0.25 / (x * x))))))))
	else:
		tmp = math.fmod((x + 1.0), 1.0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-154)
		tmp = Float64(rem(exp(x), Float64(Float64(x * x) * Float64(sqrt(0.041666666666666664) + Float64(-0.25 / Float64(Float64(x * x) * sqrt(0.041666666666666664)))))) * t_0);
	elseif (x <= -2e-310)
		tmp = Float64(t_0 * rem(exp(x), Float64(x * Float64(x * Float64(x * Float64(x * Float64(-0.010416666666666666 + Float64(-0.25 / Float64(x * x)))))))));
	else
		tmp = rem(Float64(x + 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-154], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[0.041666666666666664], $MachinePrecision] + N[(-0.25 / N[(N[(x * x), $MachinePrecision] * N[Sqrt[0.041666666666666664], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(x * N[(x * N[(-0.010416666666666666 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999997e-155

   1. Initial program 14.2%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if -9.9999999999999997e-155 < x < -1.999999999999994e-310

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. cube-mult N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 N/A

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. unpow2 N/A

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

      25. lower-*.f64 0.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{\color{blue}{x \cdot x}}\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 0.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 100.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right)\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right) \cdot x\right)}\right) \cdot e^{-x} \]

   if -1.999999999999994e-310 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 5.5

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 39.4

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

   11. Applied rewrites 39.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(\sqrt{0.041666666666666664} + \frac{-0.25}{\left(x \cdot x\right) \cdot \sqrt{0.041666666666666664}}\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 0.0)
     (*
      t_0
      (fmod
       (exp x)
       (* x (* x (* x (* x (+ -0.010416666666666666 (/ -0.25 (* x x)))))))))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (sqrt
          (fma
           (* x x)
           (fma
            (* x x)
            (fma x (* x -0.001388888888888889) 0.041666666666666664)
            -0.5)
           1.0))))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = t_0 * fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * fmod(exp(x), (x * (x * (x * (x * (-0.010416666666666666 + (-0.25 / (x * x))))))));
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), sqrt(fma((x * x), fma((x * x), fma(x, (x * -0.001388888888888889), 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(t_0 * rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_0 * rem(exp(x), Float64(x * Float64(x * Float64(x * Float64(x * Float64(-0.010416666666666666 + Float64(-0.25 / Float64(x * x)))))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), sqrt(fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.001388888888888889), 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[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(x * N[(x * N[(-0.010416666666666666 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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] * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -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 4.3%

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

   3. Taylor expanded in x around 0

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

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. cube-mult N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 N/A

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. unpow2 N/A

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

      25. lower-*.f64 1.3

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{\color{blue}{x \cdot x}}\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 1.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 28.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right)\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 28.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right) \cdot x\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 79.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. +-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right)}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. lower-*.f64 77.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right)}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 77.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}}\right)\right) \cdot e^{-x} \]

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 0.0)
     (*
      t_0
      (fmod
       (exp x)
       (* x (* x (* x (* x (+ -0.010416666666666666 (/ -0.25 (* x x)))))))))
     (if (<= t_1 2.0)
       (/
        (fmod
         (exp x)
         (fma
          x
          (*
           x
           (fma
            (* x x)
            (fma (* x x) -0.003298611111111111 -0.010416666666666666)
            -0.25))
          1.0))
        (exp x))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = t_0 * fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0 * fmod(exp(x), (x * (x * (x * (x * (-0.010416666666666666 + (-0.25 / (x * x))))))));
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(x, (x * fma((x * x), fma((x * x), -0.003298611111111111, -0.010416666666666666), -0.25)), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(t_0 * rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_0 * rem(exp(x), Float64(x * Float64(x * Float64(x * Float64(x * Float64(-0.010416666666666666 + Float64(-0.25 / Float64(x * x)))))))));
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666), -0.25)), 1.0)) / exp(x));
	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[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(x * N[(x * N[(-0.010416666666666666 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.3%

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

   3. Taylor expanded in x around 0

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

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. cube-mult N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{96} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 N/A

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. unpow2 N/A

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

      25. lower-*.f64 1.3

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{\color{blue}{x \cdot x}}\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 1.3%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right) \cdot e^{-x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 28.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)}\right)\right) \cdot x\right)\right) \cdot e^{-x} \]

   10. Applied rewrites 28.4%

       \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right) \cdot x\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 79.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right), \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-19}{5760}} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \frac{-19}{5760} + \color{blue}{\frac{-1}{96}}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-19}{5760}, \frac{-1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      16. lower-*.f64 76.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 76.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right)}\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right)\right)} + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right)} + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. lift-fmod.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right)} \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot \frac{1}{\color{blue}{e^{x}}} \]

   7. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.010416666666666666 + \frac{-0.25}{x \cdot x}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 2.0)
   (/
    (fmod
     (exp x)
     (fma
      x
      (*
       x
       (fma
        (* x x)
        (fma (* x x) -0.003298611111111111 -0.010416666666666666)
        -0.25))
      1.0))
    (exp x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((exp(-x) * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), fma(x, (x * fma((x * x), fma((x * x), -0.003298611111111111, -0.010416666666666666), -0.25)), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666), -0.25)), 1.0)) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right), \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-19}{5760}} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \frac{-19}{5760} + \color{blue}{\frac{-1}{96}}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-19}{5760}, \frac{-1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      16. lower-*.f64 8.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-19}{5760} + \frac{-1}{96}\right)\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right)}\right) + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right)\right)} + \frac{-1}{4}\right) + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right)} + 1\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. lift-fmod.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right)} \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot \frac{1}{\color{blue}{e^{x}}} \]

   7. Applied rewrites 8.9%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (*
      t_0
      (fmod
       (exp x)
       (fma
        (* x x)
        (fma
         x
         (* x (fma (* x x) -0.003298611111111111 -0.010416666666666666))
         -0.25)
        1.0)))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma(x, (x * fma((x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.003298611111111111, -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]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision]), $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 2 regimes

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

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + 1\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right), \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-19}{5760}} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \frac{-19}{5760} + \color{blue}{\frac{-1}{96}}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-19}{5760}, \frac{-1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      16. lower-*.f64 8.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 2.0)
   (/
    (fmod (exp x) (fma x (* x (fma (* x x) -0.010416666666666666 -0.25)) 1.0))
    (exp x))
   (fmod 1.0 1.0)))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 8.9

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

   5. Applied rewrites 8.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fmod.f64 N/A

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

      7. exp-neg N/A

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

      8. lift-exp.f64 N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 8.9

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

   7. Applied rewrites 8.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 27.7%

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

Alternative 7: 26.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 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 tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 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))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(Float64(x * 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]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -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 9.1%

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

   3. Taylor expanded in x around 0

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

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 8.9

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

   5. Applied rewrites 8.9%

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

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 27.7%

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

Alternative 8: 24.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.9%

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

6. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 5.4

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

8. Applied rewrites 5.4%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 26.1

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

11. Applied rewrites 26.1%

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

12. Final simplification 26.1%

    \[\leadsto \left(\left(x + 1\right) \bmod 1\right) \]
13. Add Preprocessing

Alternative 9: 23.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 24.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.1%

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

9. Taylor expanded in x around 0

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

    1. lower-fmod.f64 24.1

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

11. Applied rewrites 24.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))