expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 62.8%

Time: 14.1s

Alternatives: 11

Speedup: 4.1×

• could not determine a ground truth

  1. x = -3.8679133312993946e+136

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \frac{\frac{1}{x} + \left(0.5 + \frac{1}{x \cdot x}\right)}{x}\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.8e-103)
   (*
    (fmod
     (*
      (* x (* x x))
      (+ 0.16666666666666666 (/ (+ (/ 1.0 x) (+ 0.5 (/ 1.0 (* x x)))) x)))
     (fma x (* x -0.25) 1.0))
    (- 1.0 x))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.8e-103) {
		tmp = fmod(((x * (x * x)) * (0.16666666666666666 + (((1.0 / x) + (0.5 + (1.0 / (x * x)))) / x))), fma(x, (x * -0.25), 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.8e-103)
		tmp = Float64(rem(Float64(Float64(x * Float64(x * x)) * Float64(0.16666666666666666 + Float64(Float64(Float64(1.0 / x) + Float64(0.5 + Float64(1.0 / Float64(x * x)))) / x))), fma(x, Float64(x * -0.25), 1.0)) * Float64(1.0 - x));
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.8e-103], N[(N[With[{TMP1 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - 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 < -1.7999999999999999e-103

   1. Initial program 22.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 18.5

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

   5. Simplified 18.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 18.5

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

   8. Simplified 18.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 18.4

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

   11. Simplified 18.4%

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

   12. Taylor expanded in x around -inf

       \[\leadsto \left(\color{blue}{\left(-1 \cdot \left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left({x}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

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

          \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

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

          \[\leadsto \left(\color{blue}{\left({x}^{3} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)}{x} - \frac{1}{6}\right)\right)\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot \frac{-1}{4}, 1\right)\right)\right) \cdot \left(1 - x\right) \]

       4. cube-mult N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. distribute-neg-in N/A

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

       13. metadata-eval N/A

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

       14. mul-1-neg N/A

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

       15. remove-double-neg N/A

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

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

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

   14. Simplified 38.2%

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

   if -1.7999999999999999e-103 < x

   1. Initial program 4.8%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 4.5%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 4.5

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

   8. Simplified 4.5%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 24.0

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

   11. Simplified 24.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 69.5%

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

Alternative 2: 60.6% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (fmod
     (fma x (fma x 0.5 1.0) 1.0)
     (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0))
    (fma x (fma x 0.5 -1.0) 1.0))
   (fmod x 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), fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod(x, 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), fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 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 * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 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 10.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 10.4

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

   5. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 9.4

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

   8. Simplified 9.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 9.9

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

   11. Simplified 9.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

Alternative 3: 60.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right) \bmod 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (fmod (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0) 1.0)
    (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0), 1.0) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0), 1.0) * fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 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 * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 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 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 9.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

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

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 9.8

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

   11. Simplified 9.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

Alternative 4: 60.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0)
    (fmod (fma x (fma x 0.5 1.0) 1.0) 1.0))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0) * fmod(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0) * rem(fma(x, fma(x, 0.5, 1.0), 1.0), 1.0));
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 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], 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 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 9.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. rgt-mult-inverse N/A

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

       3. *-rgt-identity N/A

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

       4. rgt-mult-inverse N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. associate-/l* N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. *-commutative N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. associate-*r/ N/A

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

       14. distribute-lft-in N/A

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

       15. +-commutative N/A

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

       16. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 9.5%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

4. Final simplification 61.9%

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

Alternative 5: 60.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (- 1.0 x) (fmod (fma x (fma x 0.5 1.0) 1.0) (fma x (* x -0.25) 1.0)))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (1.0 - x) * fmod(fma(x, fma(x, 0.5, 1.0), 1.0), fma(x, (x * -0.25), 1.0));
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(1.0 - x) * rem(fma(x, fma(x, 0.5, 1.0), 1.0), fma(x, Float64(x * -0.25), 1.0)));
	else
		tmp = rem(x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 10.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 9.0

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

   5. Simplified 9.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 9.0

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

   8. Simplified 9.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 9.0

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

   11. Simplified 9.0%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

4. Final simplification 61.7%

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

Alternative 6: 60.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right) \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0)
    (fmod (+ x 1.0) 1.0))
   (fmod x 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0) * fmod((x + 1.0), 1.0);
	} else {
		tmp = fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0) * 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[(N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 9.9%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 8.9

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

   11. Simplified 8.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

4. Final simplification 61.7%

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

Alternative 7: 60.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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 10.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 9.0

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

   5. Simplified 9.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 9.0

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

   8. Simplified 9.0%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 8.9

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

   11. Simplified 8.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

4. Final simplification 61.7%

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

Alternative 8: 60.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], 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 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 8.1

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

   8. Simplified 8.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 8.2

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

   11. Simplified 8.2%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

Alternative 9: 59.9% accurate, 3.8× 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 10.4%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 10.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 8.1

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

   8. Simplified 8.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 8.1

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

   11. Simplified 8.1%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.6%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. Simplified 5.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. exp-lowering-exp.f64 5.1

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 33.0

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

   11. Simplified 33.0%

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

   12. Taylor expanded in x around inf

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

   14. Simplified 98.4%

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

4. Final simplification 61.3%

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

Alternative 10: 58.6% accurate, 4.1× 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 7.6%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
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. Simplified 7.3%

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

6. Taylor expanded in x around 0

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

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

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

   2. exp-lowering-exp.f64 6.3

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

8. Simplified 6.3%

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

9. Taylor expanded in x around 0

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

    1. +-lowering-+.f64 22.7

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

11. Simplified 22.7%

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

12. Taylor expanded in x around inf

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

14. Simplified 58.9%

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

Alternative 11: 23.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 20.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 20.4%

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

9. Taylor expanded in x around 0

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

    1. fmod-lowering-fmod.f64 20.4

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

11. Simplified 20.4%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))