expfmod (used to be hard to sample)

Percentage Accurate: 7.2% → 65.6%

Time: 12.8s

Alternatives: 13

Speedup: 4.1×

• could not determine a ground truth

  1. x = -4.944108507505052e+38

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: 7.2% 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: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \left(\frac{\frac{\frac{1}{x \cdot x}}{\frac{1}{x} - \mathsf{fma}\left(x, 0.5, 1\right)}}{x \cdot x} + \frac{\frac{\mathsf{fma}\left(x, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, 1\right)}{\mathsf{fma}\left(x, 0.5, 1\right) + \frac{-1}{x}}}{x \cdot x}\right)\right)\right) \bmod 1\right) \cdot t\_0\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(\frac{x \cdot x}{x}, \frac{x}{\frac{x}{1 + \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)}}, x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-72)
     (*
      (fmod
       (*
        (* x (* x x))
        (+
         0.16666666666666666
         (+
          (/ (/ (/ 1.0 (* x x)) (- (/ 1.0 x) (fma x 0.5 1.0))) (* x x))
          (/
           (/
            (* (fma x 0.5 1.0) (fma x 0.5 1.0))
            (+ (fma x 0.5 1.0) (/ -1.0 x)))
           (* x x)))))
       1.0)
      t_0)
     (if (<= x -1.55e-162)
       (*
        t_0
        (fmod
         (fma
          (/ (* x x) x)
          (/ x (/ x (+ 1.0 (fma x 0.5 (/ 1.0 x)))))
          (* x (* 0.16666666666666666 (* x x))))
         1.0))
       (if (<= x 0.6)
         (* t_0 (fmod (fma x (* x 0.5) x) 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 <= -5e-72) {
		tmp = fmod(((x * (x * x)) * (0.16666666666666666 + ((((1.0 / (x * x)) / ((1.0 / x) - fma(x, 0.5, 1.0))) / (x * x)) + (((fma(x, 0.5, 1.0) * fma(x, 0.5, 1.0)) / (fma(x, 0.5, 1.0) + (-1.0 / x))) / (x * x))))), 1.0) * t_0;
	} else if (x <= -1.55e-162) {
		tmp = t_0 * fmod(fma(((x * x) / x), (x / (x / (1.0 + fma(x, 0.5, (1.0 / x))))), (x * (0.16666666666666666 * (x * x)))), 1.0);
	} else if (x <= 0.6) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 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 <= -5e-72)
		tmp = Float64(rem(Float64(Float64(x * Float64(x * x)) * Float64(0.16666666666666666 + Float64(Float64(Float64(Float64(1.0 / Float64(x * x)) / Float64(Float64(1.0 / x) - fma(x, 0.5, 1.0))) / Float64(x * x)) + Float64(Float64(Float64(fma(x, 0.5, 1.0) * fma(x, 0.5, 1.0)) / Float64(fma(x, 0.5, 1.0) + Float64(-1.0 / x))) / Float64(x * x))))), 1.0) * t_0);
	elseif (x <= -1.55e-162)
		tmp = Float64(t_0 * rem(fma(Float64(Float64(x * x) / x), Float64(x / Float64(x / Float64(1.0 + fma(x, 0.5, Float64(1.0 / x))))), Float64(x * Float64(0.16666666666666666 * Float64(x * x)))), 1.0));
	elseif (x <= 0.6)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 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, -5e-72], N[(N[With[{TMP1 = N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x * 0.5 + 1.0), $MachinePrecision] * N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.5 + 1.0), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(t$95$0 * N[With[{TMP1 = N[(N[(N[(x * x), $MachinePrecision] / x), $MachinePrecision] * N[(x / N[(x / N[(1.0 + N[(x * 0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.6], N[(t$95$0 * N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(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 4 regimes

2. if x < -4.9999999999999996e-72

   1. Initial program 20.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 12.9

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

   8. Applied rewrites 12.9%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 27.5%

       \[\leadsto \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(0.16666666666666666 + \frac{\frac{\frac{1}{x} + \mathsf{fma}\left(x, 0.5, 1\right)}{x}}{x}\right)}\right) \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.0%

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

   if -4.9999999999999996e-72 < x < -1.5499999999999999e-162

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 7.8%

       \[\leadsto \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(0.16666666666666666 + \frac{\frac{\frac{1}{x} + \mathsf{fma}\left(x, 0.5, 1\right)}{x}}{x}\right)}\right) \bmod 1\right) \cdot e^{-x} \]
   12. Step-by-step derivation

   13. Applied rewrites 31.5%

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

   if -1.5499999999999999e-162 < x < 0.599999999999999978

   1. Initial program 5.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.9%

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

   if 0.599999999999999978 < x

   1. Initial program 2.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{\mathsf{neg}\left(x\right)} \]
   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 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \left(\color{blue}{\left(x + 1\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \left(\frac{\frac{\frac{1}{x \cdot x}}{\frac{1}{x} - \mathsf{fma}\left(x, 0.5, 1\right)}}{x \cdot x} + \frac{\frac{\mathsf{fma}\left(x, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, 1\right)}{\mathsf{fma}\left(x, 0.5, 1\right) + \frac{-1}{x}}}{x \cdot x}\right)\right)\right) \bmod 1\right) \cdot e^{-x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(\frac{x \cdot x}{x}, \frac{x}{\frac{x}{1 + \mathsf{fma}\left(x, 0.5, \frac{1}{x}\right)}}, x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right) \bmod 1\right)\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.001:\\ \;\;\;\;t\_0 \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 0.001)
     (* t_0 (fmod (fma x (* x 0.5) x) 1.0))
     (/ (fmod (+ x 1.0) 1.0) (exp x)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 0.001) {
		tmp = t_0 * fmod(fma(x, (x * 0.5), x), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 0.001)
		tmp = Float64(t_0 * rem(fma(x, Float64(x * 0.5), x), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 0.001], N[(t$95$0 * N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1e-3

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.6%

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

   if 1e-3 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 11.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.1

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

   8. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 94.2

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

   10. Applied rewrites 94.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 0.001:\\ \;\;\;\;e^{-x} \cdot \left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))