expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 62.3%

Time: 14.0s

Alternatives: 14

Speedup: 4.1×

• could not determine a ground truth

  1. x = -2.002310484613705e+242

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t\_0 \cdot e^{-x}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (* t_0 (exp (- x)))))
   (if (<= t_1 2e-9)
     (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
     (if (<= t_1 2.0) (/ t_0 (exp x)) (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = t_0 * exp(-x);
	double tmp;
	if (t_1 <= 2e-9) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else if (t_1 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = Float64(t_0 * exp(Float64(-x)))
	tmp = 0.0
	if (t_1 <= 2e-9)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{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]}, Block[{t$95$1 = N[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-9], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2.00000000000000012e-9

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 53.0

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

   11. Applied rewrites 53.0%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fmod.f64 N/A

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

       4. exp-neg N/A

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

       5. lift-exp.f64 N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 53.0

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

   13. Applied rewrites 53.0%

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

   if 2.00000000000000012e-9 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 95.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fmod.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 93.9

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

   11. Applied rewrites 93.9%

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

Alternative 2: 62.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fmod (exp x) (sqrt (cos x))) (exp (- x)))))
   (if (<= t_0 2e-9)
     (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
     (if (<= t_0 2.0)
       (/
        (fmod
         (exp x)
         (fma (* x x) (fma x (* x -0.010416666666666666) -0.25) 1.0))
        (exp x))
       (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	double tmp;
	if (t_0 <= 2e-9) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else if (t_0 <= 2.0) {
		tmp = fmod(exp(x), fma((x * x), fma(x, (x * -0.010416666666666666), -0.25), 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 2e-9)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	elseif (t_0 <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * -0.010416666666666666), -0.25), 1.0)) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-9], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2.00000000000000012e-9

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 53.0

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

   11. Applied rewrites 53.0%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fmod.f64 N/A

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

       4. exp-neg N/A

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

       5. lift-exp.f64 N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 53.0

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

   13. Applied rewrites 53.0%

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

   if 2.00000000000000012e-9 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 95.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fmod.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.0

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

   7. Applied rewrites 96.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 93.9

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

   11. Applied rewrites 93.9%

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

Alternative 3: 59.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2e-9)
   (fmod (fma x (* x 0.5) x) 1.0)
   (fmod (+ x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2e-9) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2e-9)
		tmp = rem(fma(x, Float64(x * 0.5), x), 1.0);
	else
		tmp = rem(Float64(x + 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e-9], N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2.00000000000000012e-9

   1. Initial program 5.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 53.0

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

   11. Applied rewrites 53.0%

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

   12. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-lft-identity N/A

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

       4. distribute-rgt-in N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub-inv N/A

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

       7. lower-fmod.f64 N/A

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

       8. cancel-sign-sub-inv N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

       11. distribute-lft-in N/A

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

       12. *-rgt-identity N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 53.0

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

   14. Applied rewrites 53.0%

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

   if 2.00000000000000012e-9 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 17.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 16.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 11.1

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

   8. Applied rewrites 11.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 82.5

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

   11. Applied rewrites 82.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/ (fmod (exp x) 1.0) (exp x))
   (if (<= x 400.0)
     (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-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, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fmod.f64 N/A

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

       4. exp-neg N/A

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

       5. lift-exp.f64 N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 98.3

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

   13. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 5: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (exp (- x)) (fmod (exp x) 1.0))
   (if (<= x 400.0)
     (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = exp(-x) * fmod(exp(x), 1.0);
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(exp(Float64(-x)) * rem(exp(x), 1.0));
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fmod.f64 N/A

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

       4. exp-neg N/A

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

       5. lift-exp.f64 N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 98.3

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

   13. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot \left(\left(1 - x\right) + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0) (+ (- 1.0 x) (* 0.5 (* x x))))
   (if (<= x 400.0)
     (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * ((1.0 - x) + (0.5 * (x * x)));
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * Float64(Float64(1.0 - x) + Float64(0.5 * Float64(x * x))));
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 10.4

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 10.6

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

   11. Applied rewrites 10.6%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. mul-1-neg N/A

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

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

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

       5. mul-1-neg N/A

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

       6. associate-+r+ N/A

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

   14. Applied rewrites 10.8%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fmod.f64 N/A

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

       4. exp-neg N/A

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

       5. lift-exp.f64 N/A

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

       6. un-div-inv N/A

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

       7. lower-/.f64 98.3

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

   13. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 7: 61.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot \left(\left(1 - x\right) + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0) (+ (- 1.0 x) (* 0.5 (* x x))))
   (if (<= x 400.0)
     (* (fmod (fma x (* x 0.5) x) 1.0) (fma x (fma x 0.5 -1.0) 1.0))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * ((1.0 - x) + (0.5 * (x * x)));
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * Float64(Float64(1.0 - x) + Float64(0.5 * Float64(x * x))));
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 10.4

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 10.6

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

   11. Applied rewrites 10.6%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. mul-1-neg N/A

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

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

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

       5. mul-1-neg N/A

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

       6. associate-+r+ N/A

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

   14. Applied rewrites 10.8%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 98.3

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

   14. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 8: 61.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.5 -1.0) 1.0)))
   (if (<= x -5e-310)
     (* (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0) t_0)
     (if (<= x 400.0) (* (fmod (fma x (* x 0.5) x) 1.0) t_0) (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.5, -1.0), 1.0);
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * t_0;
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * t_0;
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.5, -1.0), 1.0)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * t_0);
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * t_0);
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 10.4

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 10.7

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

   11. Applied rewrites 10.7%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 98.3

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

   14. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 9: 61.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0) (- 1.0 x))
   (if (<= x 400.0)
     (* (fmod (fma x (* x 0.5) x) 1.0) (fma x (fma x 0.5 -1.0) 1.0))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * (1.0 - x);
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * Float64(1.0 - x));
	elseif (x <= 400.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 10.4

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 10.1

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

   11. Applied rewrites 10.1%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 98.3

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

   14. Applied rewrites 98.3%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 10: 61.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0) (- 1.0 x))
   (if (<= x 1.0)
     (* (fmod (fma x (* x 0.5) x) 1.0) (- 1.0 x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * (1.0 - x);
	} else if (x <= 1.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0) * Float64(1.0 - x));
	elseif (x <= 1.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 10.4

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 10.1

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

   11. Applied rewrites 10.1%

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

   if -4.999999999999985e-310 < x < 1

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 98.3

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

   14. Applied rewrites 98.3%

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

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 11: 61.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0)
   (if (<= x 1.0)
     (* (fmod (fma x (* x 0.5) x) 1.0) (- 1.0 x))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	} else if (x <= 1.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	elseif (x <= 1.0)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 1.0], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 8.2

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. cancel-sign-sub-inv N/A

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

       4. *-commutative N/A

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

       5. cancel-sign-sub-inv N/A

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

       6. metadata-eval N/A

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

       7. lft-mult-inverse N/A

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

       8. *-lft-identity N/A

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

       9. lft-mult-inverse N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. distribute-rgt-in N/A

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

       14. +-commutative N/A

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

       15. lft-mult-inverse N/A

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

       16. distribute-rgt-in N/A

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

       17. distribute-lft-in N/A

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

   11. Applied rewrites 8.6%

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

   if -4.999999999999985e-310 < x < 1

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 98.3

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

   14. Applied rewrites 98.3%

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

   if 1 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 12: 61.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0)
   (if (<= x 400.0) (fmod (fma x (* x 0.5) x) 1.0) (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	} else if (x <= 400.0) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	elseif (x <= 400.0)
		tmp = rem(fma(x, Float64(x * 0.5), x), 1.0);
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 400.0], N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 8.2

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. cancel-sign-sub-inv N/A

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

       4. *-commutative N/A

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

       5. cancel-sign-sub-inv N/A

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

       6. metadata-eval N/A

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

       7. lft-mult-inverse N/A

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

       8. *-lft-identity N/A

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

       9. lft-mult-inverse N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. distribute-rgt-in N/A

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

       14. +-commutative N/A

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

       15. lft-mult-inverse N/A

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

       16. distribute-rgt-in N/A

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

       17. distribute-lft-in N/A

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

   11. Applied rewrites 8.6%

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

   if -4.999999999999985e-310 < x < 400

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-lft-identity N/A

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

       4. distribute-rgt-in N/A

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

       5. metadata-eval N/A

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

       6. cancel-sign-sub-inv N/A

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

       7. lower-fmod.f64 N/A

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

       8. cancel-sign-sub-inv N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

       11. distribute-lft-in N/A

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

       12. *-rgt-identity N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 98.1

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

   14. Applied rewrites 98.1%

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

   if 400 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-fmod.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 13: 24.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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 7.7%

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

6. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 6.4

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

8. Applied rewrites 6.4%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 22.8

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

11. Applied rewrites 22.8%

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

12. Final simplification 22.8%

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

Alternative 14: 23.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 21.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 20.9%

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

9. Taylor expanded in x around 0

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

    1. lower-fmod.f64 21.0

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

11. Applied rewrites 21.0%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))