expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 64.9%

Time: 13.6s

Alternatives: 14

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.5500300090886438e+276

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: 6.9% 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: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.8e-103)
     (*
      (fmod
       (*
        (* x (* x x))
        (- 0.16666666666666666 (/ (+ -0.5 (+ (/ -1.0 x) (/ -1.0 (* x x)))) x)))
       1.0)
      t_0)
     (if (<= x 0.2)
       (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
       (* t_0 (fmod 1.0 (sqrt (cos x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod(((x * (x * x)) * (0.16666666666666666 - ((-0.5 + ((-1.0 / x) + (-1.0 / (x * x)))) / x))), 1.0) * t_0;
	} else if (x <= 0.2) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = t_0 * fmod(1.0, sqrt(cos(x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64(Float64(x * Float64(x * x)) * Float64(0.16666666666666666 - Float64(Float64(-0.5 + Float64(Float64(-1.0 / x) + Float64(-1.0 / Float64(x * x)))) / x))), 1.0) * t_0);
	elseif (x <= 0.2)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = Float64(t_0 * rem(1.0, sqrt(cos(x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 - N[(N[(-0.5 + N[(N[(-1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.2], 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[(t$95$0 * N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-103

   1. Initial program 27.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 27.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 25.4

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

   8. Applied rewrites 25.4%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)} \bmod 1\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. distribute-neg-in N/A

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

       13. metadata-eval N/A

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

       14. unsub-neg N/A

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

       15. lower--.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lower-/.f64 N/A

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

   11. Applied rewrites 47.9%

       \[\leadsto \left(\color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right)} \bmod 1\right) \cdot e^{-x} \]

   if -1.7999999999999999e-103 < x < 0.20000000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.0%

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

   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 6.0

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

   8. Applied rewrites 6.0%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 63.3

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

   11. Applied rewrites 63.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   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 63.3

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

   13. Applied rewrites 63.3%

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

   if 0.20000000000000001 < x

   1. Initial program 1.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(1 \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.002:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 0.002)
   (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) * fmod(exp(x), sqrt(cos(x)))) <= 0.002) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), sqrt(cos(x)))) <= 0.002)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 0.002], 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[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 2e-3

   1. Initial program 6.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.7%

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

   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 5.7

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

   8. Applied rewrites 5.7%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 54.0

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

   11. Applied rewrites 54.0%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   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 54.0

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

   13. Applied rewrites 54.0%

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

   if 2e-3 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 14.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 14.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 92.1

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

   8. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 92.1

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

   10. Applied rewrites 92.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.002:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.002:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 0.002)
     (* t_0 (fmod (fma x (* x 0.5) x) 1.0))
     (/ (fmod (+ x 1.0) 1.0) (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 0.002) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 0.002)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 0.002], N[(t$95$0 * N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 2e-3

   1. Initial program 6.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 5.7%

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

   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 5.7

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

   8. Applied rewrites 5.7%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 54.0

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

   11. Applied rewrites 54.0%

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

   if 2e-3 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 14.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 14.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 92.1

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

   8. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 92.1

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

   10. Applied rewrites 92.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.002:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (exp (- x)) (fmod (exp x) (sqrt (cos x)))) 5e-11)
   (* (fmod (fma x (* x 0.5) x) 1.0) (fma x (fma x 0.5 -1.0) 1.0))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) * fmod(exp(x), sqrt(cos(x)))) <= 5e-11) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), sqrt(cos(x)))) <= 5e-11)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 5e-11], 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[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 5.00000000000000018e-11

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.0%

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

   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 5.1

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

   8. Applied rewrites 5.1%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 54.4

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

   11. Applied rewrites 54.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 54.4%

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

   if 5.00000000000000018e-11 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 18.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 16.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 88.9

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

   8. Applied rewrites 88.9%

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

      1. lift-+.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 5e-11)
     (* (fmod (fma x (* x 0.5) x) 1.0) (fma x (fma x 0.5 -1.0) 1.0))
     (* t_0 (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 5e-11) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = t_0 * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 5e-11)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 5e-11], 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[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 5.00000000000000018e-11

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.0%

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

   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 5.1

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

   8. Applied rewrites 5.1%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 54.4

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

   11. Applied rewrites 54.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 54.4%

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

   if 5.00000000000000018e-11 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 18.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 16.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.8e-103)
     (*
      (fmod
       (*
        (* x (* x x))
        (- 0.16666666666666666 (/ (+ -0.5 (+ (/ -1.0 x) (/ -1.0 (* x x)))) x)))
       1.0)
      t_0)
     (if (<= x 0.2)
       (/ (fmod (fma x (* x 0.5) x) 1.0) (exp x))
       (* t_0 (fmod (+ x 1.0) 1.0))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod(((x * (x * x)) * (0.16666666666666666 - ((-0.5 + ((-1.0 / x) + (-1.0 / (x * x)))) / x))), 1.0) * t_0;
	} else if (x <= 0.2) {
		tmp = fmod(fma(x, (x * 0.5), x), 1.0) / exp(x);
	} else {
		tmp = t_0 * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64(Float64(x * Float64(x * x)) * Float64(0.16666666666666666 - Float64(Float64(-0.5 + Float64(Float64(-1.0 / x) + Float64(-1.0 / Float64(x * x)))) / x))), 1.0) * t_0);
	elseif (x <= 0.2)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), 1.0) / exp(x));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 - N[(N[(-0.5 + N[(N[(-1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.2], 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[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e-103

   1. Initial program 27.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 27.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 25.4

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

   8. Applied rewrites 25.4%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)} \bmod 1\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. distribute-neg-in N/A

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

       13. metadata-eval N/A

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

       14. unsub-neg N/A

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

       15. lower--.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lower-/.f64 N/A

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

   11. Applied rewrites 47.9%

       \[\leadsto \left(\color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right)} \bmod 1\right) \cdot e^{-x} \]

   if -1.7999999999999999e-103 < x < 0.20000000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.0%

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

   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 6.0

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

   8. Applied rewrites 6.0%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 63.3

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

   11. Applied rewrites 63.3%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod 1\right) \cdot e^{-x} \]
   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 63.3

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

   13. Applied rewrites 63.3%

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

   if 0.20000000000000001 < x

   1. Initial program 1.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 97.1

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

   8. Applied rewrites 97.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 - \frac{-0.5 + \left(\frac{-1}{x} + \frac{-1}{x \cdot x}\right)}{x}\right)\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.5 -1.0) 1.0)))
   (if (<= x -5e-310)
     (* t_0 (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0))
     (if (<= x 200.0) (* (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 = t_0 * fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	} else if (x <= 200.0) {
		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(t_0 * rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0));
	elseif (x <= 200.0)
		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[(t$95$0 * 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]), $MachinePrecision], If[LessEqual[x, 200.0], 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 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 12.9%

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

   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 11.8

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

   8. Applied rewrites 11.8%

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

   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 12.4

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

   11. Applied rewrites 12.4%

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

   if -4.999999999999985e-310 < x < 200

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.8%

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

   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 7.9

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

   8. Applied rewrites 7.9%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 96.3

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

   11. Applied rewrites 96.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 96.1%

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

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 64.9%

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

Alternative 8: 62.3% accurate, 3.0× speedup

Math

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

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 200.0)
     (* (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 <= 200.0) {
		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 <= 200.0)
		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, 200.0], 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 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 12.9%

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

   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 11.8

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

   8. Applied rewrites 11.8%

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

   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 10.9

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

   11. Applied rewrites 10.9%

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

   if -4.999999999999985e-310 < x < 200

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.8%

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

   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 7.9

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

   8. Applied rewrites 7.9%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 96.3

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

   11. Applied rewrites 96.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 96.1%

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

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 9: 62.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

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 0.2)
     (* (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 <= 0.2) {
		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 <= 0.2)
		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, 0.2], 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 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 12.9%

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

   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 11.8

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

   8. Applied rewrites 11.8%

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

   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 10.9

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

   11. Applied rewrites 10.9%

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

   if -4.999999999999985e-310 < x < 0.20000000000000001

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0

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

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \bmod 1\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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 7.7

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

   8. Applied rewrites 7.7%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 97.9

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

   11. Applied rewrites 97.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 97.9%

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

   if 0.20000000000000001 < x

   1. Initial program 1.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.3

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.6%

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

Alternative 10: 62.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (+ x 1.0) 1.0) (- 1.0 x))
   (if (<= x 0.2)
     (* (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((x + 1.0), 1.0) * (1.0 - x);
	} else if (x <= 0.2) {
		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(Float64(x + 1.0), 1.0) * Float64(1.0 - x));
	elseif (x <= 0.2)
		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 + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.2], 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 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 12.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-lft1-in N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-fmod.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. neg-mul-1 N/A

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

       9. unsub-neg N/A

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

       10. lower--.f64 10.5

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

   11. Applied rewrites 10.5%

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

   if -4.999999999999985e-310 < x < 0.20000000000000001

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0

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

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \bmod 1\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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 7.7

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

   8. Applied rewrites 7.7%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 97.9

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

   11. Applied rewrites 97.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 97.9%

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

   if 0.20000000000000001 < x

   1. Initial program 1.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.3

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (fmod (+ x 1.0) 1.0) (- 1.0 x))
   (if (<= x 200.0) (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((x + 1.0), 1.0) * (1.0 - x);
	} else if (x <= 200.0) {
		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 = Float64(rem(Float64(x + 1.0), 1.0) * Float64(1.0 - x));
	elseif (x <= 200.0)
		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[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], 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 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 12.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-lft1-in N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-fmod.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. neg-mul-1 N/A

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

       9. unsub-neg N/A

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

       10. lower--.f64 10.5

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

   11. Applied rewrites 10.5%

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

   if -4.999999999999985e-310 < x < 200

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 7.8%

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

   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 7.9

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

   8. Applied rewrites 7.9%

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

   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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       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 96.3

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

   11. Applied rewrites 96.3%

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

   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. *-rgt-identity N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-lft-identity N/A

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

       7. lft-mult-inverse N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lft-mult-inverse N/A

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

       12. distribute-rgt-in N/A

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

       13. +-commutative N/A

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

       14. associate-*l* N/A

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

       15. unpow2 N/A

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

       16. cube-mult N/A

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

   14. Applied rewrites 96.0%

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

   if 200 < x

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 64.1%

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

Alternative 12: 25.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 8.1%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 28.3

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. distribute-lft1-in N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower-fmod.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. neg-mul-1 N/A

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

    9. unsub-neg N/A

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

    10. lower--.f64 26.6

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

11. Applied rewrites 26.6%

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

12. Final simplification 26.6%

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

Alternative 13: 24.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 8.1%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 28.3

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in x around 0

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

    1. lower-fmod.f64 N/A

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

    2. lower-+.f64 25.8

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

11. Applied rewrites 25.8%

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

12. Final simplification 25.8%

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

Alternative 14: 23.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 8.1%

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

6. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 6.2

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

8. Applied rewrites 6.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 24.4%

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

Reproduce

herbie shell --seed 2024212 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))