expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 65.5%

Time: 14.4s

Alternatives: 6

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.0745699522729976e+234

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - x}\\ t_1 := x \cdot \mathsf{fma}\left(x, x, 0\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(t\_1 \cdot \frac{t\_1}{\mathsf{fma}\left(x, t\_1, 0\right)}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 x))) (t_1 (* x (fma x x 0.0))))
   (if (<= x -1.35e-108)
     (*
      (fmod
       (exp x)
       (fma
        (fma x x 0.0)
        (fma
         x
         (*
          x
          (fma
           (* t_1 (/ t_1 (fma x t_1 0.0)))
           -0.003298611111111111
           -0.010416666666666666))
         -0.25)
        1.0))
      t_0)
     (* t_0 (fmod x 1.0)))))

C

double code(double x) {
	double t_0 = exp((0.0 - x));
	double t_1 = x * fma(x, x, 0.0);
	double tmp;
	if (x <= -1.35e-108) {
		tmp = fmod(exp(x), fma(fma(x, x, 0.0), fma(x, (x * fma((t_1 * (t_1 / fma(x, t_1, 0.0))), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0)) * t_0;
	} else {
		tmp = t_0 * fmod(x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(0.0 - x))
	t_1 = Float64(x * fma(x, x, 0.0))
	tmp = 0.0
	if (x <= -1.35e-108)
		tmp = Float64(rem(exp(x), fma(fma(x, x, 0.0), fma(x, Float64(x * fma(Float64(t_1 * Float64(t_1 / fma(x, t_1, 0.0))), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0)) * t_0);
	else
		tmp = Float64(t_0 * rem(x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-108], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * N[(N[(t$95$1 * N[(t$95$1 / N[(x * t$95$1 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000002e-108

   1. Initial program 15.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-lft-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{0 + {x}^{2}}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

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

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, {x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)\right) + \color{blue}{\frac{-1}{4}}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right), \frac{-1}{4}\right)}, 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. sub-neg N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-19}{5760} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-19}{5760}} + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right)\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \frac{-19}{5760} + \color{blue}{\frac{-1}{96}}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

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

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-19}{5760}, \frac{-1}{96}\right)}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      17. +-lft-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{0 + {x}^{2}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      18. +-commutative N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      19. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      20. accelerator-lowering-fma.f64 15.8

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   5. Simplified 15.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)}\right) \cdot e^{-x} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x - 0}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \color{blue}{0}}{x \cdot x - 0}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      3. --rgt-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x - 0}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}{x \cdot x - 0}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      5. unpow3 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{\color{blue}{{x}^{3}} \cdot x}{x \cdot x - 0}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      6. flip-- N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x + 0}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \color{blue}{0}}{x \cdot x + 0}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x + 0}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      9. +-rgt-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{x \cdot x}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      10. div-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{x \cdot x}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      11. +-rgt-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{x \cdot x + 0}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      12. mul0-lft N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{x \cdot x + \color{blue}{0 \cdot x}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      13. +-lft-identity N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{0 + \left(x \cdot x + 0 \cdot x\right)}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3} \cdot x}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(x \cdot x + 0 \cdot x\right)}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      15. times-frac N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{\frac{{x}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{x}{\frac{1}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

      16. un-div-inv N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\frac{{x}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{1}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}\right)}, \frac{-19}{5760}, \frac{-1}{96}\right), \frac{-1}{4}\right), 1\right)\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]

   7. Applied egg-rr 38.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x, 0\right), 0\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{-x} \]

   if -1.35000000000000002e-108 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 28.5

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

   8. Simplified 28.5%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 71.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right) \cdot \frac{x \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x, 0\right), 0\right)}, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - x} \cdot \left(x \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - x} \cdot \left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/ (fmod (exp x) 1.0) (exp x))
   (* (exp (- 0.0 x)) (fmod x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = exp((0.0 - x)) * 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(exp(x), 1.0d0) / exp(x)
    else
        tmp = exp((0.0d0 - x)) * mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.exp((0.0 - x)) * math.fmod(x, 1.0)
	return tmp

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] * N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 7.3%

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

      1. exp-neg N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. exp-lowering-exp.f64 7.3

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

   7. Applied egg-rr 7.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 38.1

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 97.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - x} \cdot \left(x \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 x))))
   (if (<= x -5e-310) (* t_0 (fmod (exp x) 1.0)) (* t_0 (fmod x 1.0)))))

C

double code(double x) {
	double t_0 = exp((0.0 - x));
	double tmp;
	if (x <= -5e-310) {
		tmp = t_0 * fmod(exp(x), 1.0);
	} else {
		tmp = t_0 * fmod(x, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - x))
    if (x <= (-5d-310)) then
        tmp = t_0 * mod(exp(x), 1.0d0)
    else
        tmp = t_0 * mod(x, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp((0.0 - x))
	tmp = 0
	if x <= -5e-310:
		tmp = t_0 * math.fmod(math.exp(x), 1.0)
	else:
		tmp = t_0 * math.fmod(x, 1.0)
	return tmp

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 7.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 38.1

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 97.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{0 - x} \cdot \left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{0 - x} \cdot \left(x \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{0 - x} \cdot \left(x \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (exp (- 0.0 x)) (fmod x 1.0)))

C

double code(double x) {
	return exp((0.0 - x)) * fmod(x, 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((0.0d0 - x)) * mod(x, 1.0d0)
end function

Python

def code(x):
	return math.exp((0.0 - x)) * math.fmod(x, 1.0)

Julia

function code(x)
	return Float64(exp(Float64(0.0 - x)) * rem(x, 1.0))
end

Wolfram

code[x_] := N[(N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] * N[With[{TMP1 = x, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 6.8%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 26.6

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

8. Simplified 26.6%

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

9. Taylor expanded in x around inf

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

11. Simplified 63.1%

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

12. Final simplification 63.1%

    \[\leadsto e^{0 - x} \cdot \left(x \bmod 1\right) \]
13. Add Preprocessing

Alternative 5: 58.8% 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 6.8%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 26.6

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

8. Simplified 26.6%

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

9. Taylor expanded in x around inf

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

11. Simplified 63.1%

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

12. Taylor expanded in x around 0

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

    1. fmod-lowering-fmod.f64 62.1

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

14. Simplified 62.1%

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

Alternative 6: 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

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

5. Simplified 6.8%

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

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 5.9

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

8. Simplified 5.9%

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

9. Taylor expanded in x around 0

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

11. Simplified 24.0%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))