expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 63.1%

Time: 12.8s

Alternatives: 8

Speedup: 3.9×

• could not determine a ground truth

  1. x = -1.7216736754739248e+239

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.003298611111111111, -0.010416666666666666 + \frac{-0.25}{x \cdot x}\right), 1\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.6e-162)
     (*
      (fmod
       (exp x)
       (fma
        (* x x)
        (*
         (* x x)
         (fma
          x
          (* x -0.003298611111111111)
          (+ -0.010416666666666666 (/ -0.25 (* x x)))))
        1.0))
      t_0)
     (if (<= x 0.88)
       (*
        (fmod
         (fma (* x x) (fma x 0.16666666666666666 0.5) x)
         (fma (* x x) (fma x (* x -0.010416666666666666) -0.25) 1.0))
        (fma x (fma x 0.5 -1.0) 1.0))
       (* t_0 (fmod (+ x 1.0) (sqrt (cos x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.6e-162) {
		tmp = fmod(exp(x), fma((x * x), ((x * x) * fma(x, (x * -0.003298611111111111), (-0.010416666666666666 + (-0.25 / (x * x))))), 1.0)) * t_0;
	} else if (x <= 0.88) {
		tmp = fmod(fma((x * x), fma(x, 0.16666666666666666, 0.5), x), fma((x * x), fma(x, (x * -0.010416666666666666), -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = t_0 * fmod((x + 1.0), sqrt(cos(x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.6e-162)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), Float64(Float64(x * x) * fma(x, Float64(x * -0.003298611111111111), Float64(-0.010416666666666666 + Float64(-0.25 / Float64(x * x))))), 1.0)) * t_0);
	elseif (x <= 0.88)
		tmp = Float64(rem(fma(Float64(x * x), fma(x, 0.16666666666666666, 0.5), x), fma(Float64(x * x), fma(x, Float64(x * -0.010416666666666666), -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), sqrt(cos(x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.6e-162], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.003298611111111111), $MachinePrecision] + N[(-0.010416666666666666 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999988e-162

   1. Initial program 12.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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. lower-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. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 12.6

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

   5. Applied rewrites 12.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 14.5%

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

   if -1.59999999999999988e-162 < x < 0.880000000000000004

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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. lower-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. lower-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 6.1

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

   8. Applied rewrites 6.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       7. lower-fma.f64 6.1

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   11. Applied rewrites 6.1%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 60.1%

       \[\leadsto \left(\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   if 0.880000000000000004 < x

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.003298611111111111, -0.010416666666666666 + \frac{-0.25}{x \cdot x}\right), 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* t_0 (fmod (exp x) (sqrt (cos x))))))
   (if (<= t_1 1e-9)
     (*
      (fmod
       (fma (* x x) (fma x 0.16666666666666666 0.5) x)
       (fma (* x x) (fma x (* x -0.010416666666666666) -0.25) 1.0))
      (fma x (fma x 0.5 -1.0) 1.0))
     (if (<= t_1 2.0)
       (*
        t_0
        (fmod
         (exp x)
         (fma
          (* x x)
          (fma
           x
           (* x (fma (* x x) -0.003298611111111111 -0.010416666666666666))
           -0.25)
          1.0)))
       (* (fmod 1.0 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = t_0 * fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (t_1 <= 1e-9) {
		tmp = fmod(fma((x * x), fma(x, 0.16666666666666666, 0.5), x), fma((x * x), fma(x, (x * -0.010416666666666666), -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else if (t_1 <= 2.0) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma(x, (x * fma((x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0));
	} else {
		tmp = fmod(1.0, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(t_0 * rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (t_1 <= 1e-9)
		tmp = Float64(rem(fma(Float64(x * x), fma(x, 0.16666666666666666, 0.5), x), fma(Float64(x * x), fma(x, Float64(x * -0.010416666666666666), -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0));
	elseif (t_1 <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0)));
	else
		tmp = Float64(rem(1.0, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-9], N[(N[With[{TMP1 = N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1.00000000000000006e-9

   1. Initial program 4.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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. lower-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. lower-*.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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 4.7

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

   8. Applied rewrites 4.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]

       7. lower-fma.f64 4.7

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   11. Applied rewrites 4.7%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \left(\left({x}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{-1}{96}, \frac{-1}{4}\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{2}, -1\right), 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 51.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   if 1.00000000000000006e-9 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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. lower-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. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 97.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-9}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x\right)\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.010416666666666666, -0.25\right), 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))