expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 61.7%

Time: 11.4s

Alternatives: 14

Speedup: 3.9×

• could not determine a ground truth

  1. x = -3.2022393250915237e+187

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (fmod (exp x) (sqrt (cos x))))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_2 2.0) (/ t_1 (exp x)) (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x)));
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_2 <= 2.0) {
		tmp = t_1 / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = rem(exp(x), sqrt(cos(x)))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_2 <= 2.0)
		tmp = Float64(t_1 / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$1 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

   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

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_1 2.0) t_1 (* (fmod 1.0 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 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = fmod(1.0, 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 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(rem(1.0, 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, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], t$95$1, N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_1 2.0)
       (/
        (fmod
         (exp x)
         (fma
          (fma
           (fma -0.003298611111111111 (* x x) -0.010416666666666666)
           (* x x)
           -0.25)
          (* x x)
          1.0))
        (exp x))
       (* (fmod 1.0 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 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(fma(fma(-0.003298611111111111, (x * x), -0.010416666666666666), (x * x), -0.25), (x * x), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, 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 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)) / exp(x));
	else
		tmp = Float64(rem(1.0, 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, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

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

   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

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_1 2.0)
       (*
        (fmod
         (exp x)
         (fma
          (fma
           (fma -0.003298611111111111 (* x x) -0.010416666666666666)
           (* x x)
           -0.25)
          (* x x)
          1.0))
        t_0)
       (* (fmod 1.0 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 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(fma(fma(-0.003298611111111111, (x * x), -0.010416666666666666), (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, 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 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, 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, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_1 2.0)
       (/
        (fmod
         (exp x)
         (fma (fma (* x x) -0.010416666666666666 -0.25) (* x x) 1.0))
        (exp x))
       (* (fmod 1.0 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 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(fma((x * x), -0.010416666666666666, -0.25), (x * x), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, 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 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(Float64(x * x), -0.010416666666666666, -0.25), Float64(x * x), 1.0)) / exp(x));
	else
		tmp = Float64(rem(1.0, 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, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 87.5

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

   7. Applied rewrites 87.5%

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

   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

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), x \cdot x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\\ t_1 := e^{-x}\\ t_2 := t\_1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
        (t_1 (exp (- x)))
        (t_2 (* t_1 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_2 1e-8)
     (* (fmod (* (fma 0.5 x 1.0) x) t_0) t_1)
     (if (<= t_2 2.0) (* (fmod (exp x) t_0) t_1) (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0);
	double t_1 = exp(-x);
	double t_2 = t_1 * fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (t_2 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), t_0) * t_1;
	} else if (t_2 <= 2.0) {
		tmp = fmod(exp(x), t_0) * t_1;
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_1 * rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (t_2 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), t_0) * t_1);
	elseif (t_2 <= 2.0)
		tmp = Float64(rem(exp(x), t_0) * t_1);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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$2, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-8)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (if (<= t_1 2.0)
       (/
        (fmod
         (exp x)
         (fma
          (fma
           (fma -0.003298611111111111 (* x x) -0.010416666666666666)
           (* x x)
           -0.25)
          (* x x)
          1.0))
        (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
       (* (fmod 1.0 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 <= 1e-8) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else if (t_1 <= 2.0) {
		tmp = fmod(exp(x), fma(fma(fma(-0.003298611111111111, (x * x), -0.010416666666666666), (x * x), -0.25), (x * x), 1.0)) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, 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 <= 1e-8)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	elseif (t_1 <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, 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, 1e-8], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $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))) < 1e-8

   1. Initial program 5.4%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 5.4

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 5.4%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 1e-8 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760}, x \cdot x, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)} \]

      8. lower-fma.f64 72.7

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)} \]

   10. Applied rewrites 72.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-8}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111, x \cdot x, -0.010416666666666666\right), x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (/ (fmod (exp x) 1.0) (exp x))
   (if (<= x 200.0)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      (exp (- x)))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else if (x <= 200.0) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * exp(-x);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	elseif (x <= 200.0)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 10.0

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

   7. Applied rewrites 10.0%

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

   if -1.999999999999994e-310 < x < 200

   1. Initial program 11.8%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 10.3

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 10.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 95.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-310)
     (* (fmod (exp x) 1.0) t_0)
     (if (<= x 200.0)
       (*
        (fmod
         (* (fma 0.5 x 1.0) x)
         (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
        t_0)
       (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else if (x <= 200.0) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	elseif (x <= 200.0)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 200.0], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.0%

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

   if -1.999999999999994e-310 < x < 200

   1. Initial program 11.8%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 10.3

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 10.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 95.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (/
    (fmod (exp x) 1.0)
    (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
   (if (<= x 200.0)
     (*
      (fmod
       (* (fma 0.5 x 1.0) x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      (exp (- x)))
     (* (fmod 1.0 1.0) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = fmod(exp(x), 1.0) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else if (x <= 200.0) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * exp(-x);
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(rem(exp(x), 1.0) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	elseif (x <= 200.0)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * exp(Float64(-x)));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 10.0

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

   7. Applied rewrites 10.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 8.5

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

   10. Applied rewrites 8.5%

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

   if -1.999999999999994e-310 < x < 200

   1. Initial program 11.8%

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

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 10.3

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}, x, 1\right)\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   8. Applied rewrites 10.3%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 95.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-fmod.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-fmod.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-cos.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 24.8%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 24.8%

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

Alternative 12: 24.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-fmod.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-cos.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 24.8%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
12. Add Preprocessing

Alternative 13: 23.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-fmod.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-cos.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 24.8%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 24.2%

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

15. Final simplification 24.2%

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

Alternative 14: 22.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-fmod.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-cos.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 6.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 22.0%

    \[\leadsto \left(1 - x\right) \cdot \left(1 \bmod 1\right) \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 22.0%

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

15. Final simplification 22.0%

    \[\leadsto \left(1 \bmod 1\right) \cdot 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))