expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 61.7%

Time: 19.3s

Alternatives: 6

Speedup: 5.0×

• could not determine a ground truth

  1. x = -3.1885055540104476e+89

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: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(-0.001388888888888889 + \frac{0.041666666666666664}{x \cdot x}\right)\right)\right)\right)\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55e-162)
   (/
    (fmod
     (exp x)
     (sqrt
      (+
       1.0
       (*
        x
        (*
         x
         (+
          -0.5
          (*
           (* x x)
           (*
            x
            (*
             x
             (+
              -0.001388888888888889
              (/ 0.041666666666666664 (* x x))))))))))))
    (exp x))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.55e-162) {
		tmp = fmod(exp(x), sqrt((1.0 + (x * (x * (-0.5 + ((x * x) * (x * (x * (-0.001388888888888889 + (0.041666666666666664 / (x * x)))))))))))) / exp(x);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d-162)) then
        tmp = mod(exp(x), sqrt((1.0d0 + (x * (x * ((-0.5d0) + ((x * x) * (x * (x * ((-0.001388888888888889d0) + (0.041666666666666664d0 / (x * x)))))))))))) / exp(x)
    else
        tmp = mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1.55e-162:
		tmp = math.fmod(math.exp(x), math.sqrt((1.0 + (x * (x * (-0.5 + ((x * x) * (x * (x * (-0.001388888888888889 + (0.041666666666666664 / (x * x)))))))))))) / math.exp(x)
	else:
		tmp = math.fmod(x, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55e-162)
		tmp = Float64(rem(exp(x), sqrt(Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(-0.001388888888888889 + Float64(0.041666666666666664 / Float64(x * x)))))))))))) / exp(x));
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(-0.001388888888888889 + N[(0.041666666666666664 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5499999999999999e-162

   1. Initial program 15.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 15.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 15.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      17. *-lowering-*.f64 15.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   7. Simplified 15.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right)\right)}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} + \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \left(\frac{\frac{1}{24} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \left(\frac{\frac{1}{24}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \mathsf{/.f64}\left(\frac{1}{24}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \mathsf{/.f64}\left(\frac{1}{24}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      13. *-lowering-*.f64 17.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{720}, \mathsf{/.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   10. Simplified 17.0%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 5.4%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 5.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 5.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 4.5%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 4.5%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 4.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 26.3%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 26.3%

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

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{x}, 1\right) \]
   15. Step-by-step derivation

   16. Simplified 73.9%

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

Alternative 2: 61.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(1 + x \cdot \left(x \cdot \left(-0.25 + x \cdot \left(x \cdot -0.010416666666666666\right)\right)\right)\right)\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/
    (fmod
     (+ x 1.0)
     (+ 1.0 (* x (* x (+ -0.25 (* x (* x -0.010416666666666666)))))))
    (+ x 1.0))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod((x + 1.0), (1.0 + (x * (x * (-0.25 + (x * (x * -0.010416666666666666))))))) / (x + 1.0);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = mod((x + 1.0d0), (1.0d0 + (x * (x * ((-0.25d0) + (x * (x * (-0.010416666666666666d0)))))))) / (x + 1.0d0)
    else
        tmp = mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod((x + 1.0), (1.0 + (x * (x * (-0.25 + (x * (x * -0.010416666666666666))))))) / (x + 1.0)
	else:
		tmp = math.fmod(x, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(Float64(x + 1.0), Float64(1.0 + Float64(x * Float64(x * Float64(-0.25 + Float64(x * Float64(x * -0.010416666666666666))))))) / Float64(x + 1.0));
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(x * N[(x * N[(-0.25 + N[(x * N[(x * -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 8.9%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 8.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 8.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{96} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{4}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{4} + \frac{-1}{96} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\frac{-1}{96} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \left({x}^{2} \cdot \frac{-1}{96}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(x \cdot x\right) \cdot \frac{-1}{96}\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \left(x \cdot \left(x \cdot \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

      14. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + x\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 8.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{x}\right)\right) \]

   10. Simplified 8.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, x\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 8.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{96}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, x\right)\right) \]

   13. Simplified 8.9%

       \[\leadsto \frac{\left(\color{blue}{\left(1 + x\right)} \bmod \left(1 + x \cdot \left(x \cdot \left(-0.25 + x \cdot \left(x \cdot -0.010416666666666666\right)\right)\right)\right)\right)}{1 + x} \]

   if -4.999999999999985e-310 < x

   1. Initial program 6.1%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 6.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 6.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 5.0%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 4.9%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 4.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 33.8%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 33.8%

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

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{x}, 1\right) \]
   15. Step-by-step derivation

   16. Simplified 97.2%

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

4. Final simplification 62.7%

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

Alternative 3: 60.1% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = mod((1.0d0 + (x * (1.0d0 + (x * 0.5d0)))), 1.0d0)
    else
        tmp = mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod((1.0 + (x * (1.0 + (x * 0.5)))), 1.0)
	else:
		tmp = math.fmod(x, 1.0)
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 8.9%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 8.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 8.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 6.6%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 6.6%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right), 1\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right), 1\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right), 1\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 6.6%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), 1\right) \]

   13. Simplified 6.6%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.1%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 6.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 6.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 5.0%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 4.9%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 4.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 33.8%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 33.8%

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

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{x}, 1\right) \]
   15. Step-by-step derivation

   16. Simplified 97.2%

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

Alternative 4: 60.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = mod((x + 1.0d0), 1.0d0)
    else
        tmp = mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod((x + 1.0), 1.0)
	else:
		tmp = math.fmod(x, 1.0)
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 8.9%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 8.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 8.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 6.6%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 6.6%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 6.6%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 6.6%

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

   if -4.999999999999985e-310 < x

   1. Initial program 6.1%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 6.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 6.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 5.0%

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

   8. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

      2. exp-lowering-exp.f64 4.9%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

   10. Simplified 4.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 33.8%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 33.8%

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

   14. Taylor expanded in x around inf

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{x}, 1\right) \]
   15. Step-by-step derivation

   16. Simplified 97.2%

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

4. Final simplification 61.8%

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

Alternative 5: 58.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

   5. fmod-lowering-fmod.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

   9. exp-lowering-exp.f64 7.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

3. Simplified 7.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
6. Step-by-step derivation

7. Simplified 6.5%

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

8. Taylor expanded in x around 0

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

   1. fmod-lowering-fmod.f64 N/A

      \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{1}\right) \]

   2. exp-lowering-exp.f64 5.6%

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), 1\right) \]

10. Simplified 5.6%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
12. Step-by-step derivation

    1. +-lowering-+.f64 23.2%

       \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

13. Simplified 23.2%

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

14. Taylor expanded in x around inf

    \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{x}, 1\right) \]
15. Step-by-step derivation

16. Simplified 60.1%

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

Alternative 6: 22.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

   5. fmod-lowering-fmod.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

   9. exp-lowering-exp.f64 7.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

3. Simplified 7.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]
6. Step-by-step derivation

7. Simplified 22.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(1, \color{blue}{1}\right), \mathsf{exp.f64}\left(x\right)\right) \]
9. Step-by-step derivation

10. Simplified 21.9%

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

11. Taylor expanded in x around 0

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

    1. fmod-lowering-fmod.f64 21.9%

       \[\leadsto \mathsf{fmod.f64}\left(1, \color{blue}{1}\right) \]

13. Simplified 21.9%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))