expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 99.2%

Time: 9.9s

Alternatives: 3

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.418286241495677e+161

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;{\left({\left(e^{x \cdot x - {t\_0}^{2}}\right)}^{\left({\left(t\_0 + x\right)}^{-1}\right)}\right)}^{-1}\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (fma (* -0.25 x) x 1.0)))) (t_1 (exp (- x))))
   (if (<= x -1e-309)
     (pow (pow (exp (- (* x x) (pow t_0 2.0))) (pow (+ t_0 x) -1.0)) -1.0)
     (if (<= x 0.2)
       (* (fmod (* (fma 0.5 x 1.0) x) (fma -0.25 (* x x) 1.0)) t_1)
       t_1))))

C

double code(double x) {
	double t_0 = log(fmod(exp(x), fma((-0.25 * x), x, 1.0)));
	double t_1 = exp(-x);
	double tmp;
	if (x <= -1e-309) {
		tmp = pow(pow(exp(((x * x) - pow(t_0, 2.0))), pow((t_0 + x), -1.0)), -1.0);
	} else if (x <= 0.2) {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(-0.25, (x * x), 1.0)) * t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = log(rem(exp(x), fma(Float64(-0.25 * x), x, 1.0)))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1e-309)
		tmp = (exp(Float64(Float64(x * x) - (t_0 ^ 2.0))) ^ (Float64(t_0 + x) ^ -1.0)) ^ -1.0;
	elseif (x <= 0.2)
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(-0.25, Float64(x * x), 1.0)) * t_1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.25 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1e-309], N[Power[N[Power[N[Exp[N[(N[(x * x), $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$0 + x), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[x, 0.2], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$1), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 10.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 10.6

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

   4. Applied rewrites 10.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied rewrites 10.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 10.6

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

   9. Applied rewrites 10.6%

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

       1. lift-/.f64 N/A

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

       2. rem-exp-log N/A

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

       3. lift-log.f64 N/A

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

       4. lift-exp.f64 N/A

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

       5. exp-diff N/A

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

       6. flip-- N/A

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

       7. div-inv N/A

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

       8. exp-prod N/A

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\color{blue}{{\left(e^{x \cdot x - {\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)}^{2}}\right)}^{\left({\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right) + x\right)}^{-1}\right)}}} \]

   if -1.000000000000002e-309 < x < 0.20000000000000001

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 7.7

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

   11. Applied rewrites 7.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.2%

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

   if 0.20000000000000001 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 0.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   11. Step-by-step derivation

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 100.0

          \[\leadsto e^{\color{blue}{-x}} \]

   12. Applied rewrites 100.0%

       \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;{\left({\left(e^{x \cdot x - {\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right)}^{2}}\right)}^{\left({\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25 \cdot x, x, 1\right)\right)\right) + x\right)}^{-1}\right)}\right)}^{-1}\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-309} \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (or (<= x -1e-309) (not (<= x 0.2)))
     t_0
     (* (fmod (* (fma 0.5 x 1.0) x) (fma -0.25 (* x x) 1.0)) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((x <= -1e-309) || !(x <= 0.2)) {
		tmp = t_0;
	} else {
		tmp = fmod((fma(0.5, x, 1.0) * x), fma(-0.25, (x * x), 1.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if ((x <= -1e-309) || !(x <= 0.2))
		tmp = t_0;
	else
		tmp = Float64(rem(Float64(fma(0.5, x, 1.0) * x), fma(-0.25, Float64(x * x), 1.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[Or[LessEqual[x, -1e-309], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], t$95$0, N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309 or 0.20000000000000001 < x

   1. Initial program 7.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 7.0

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   7. Applied rewrites 7.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   11. Step-by-step derivation

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 95.8

          \[\leadsto e^{\color{blue}{-x}} \]

   12. Applied rewrites 95.8%

       \[\leadsto e^{\color{blue}{-x}} \]

   if -1.000000000000002e-309 < x < 0.20000000000000001

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 7.7

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

   11. Applied rewrites 7.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 98.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot \color{blue}{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309} \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

code[x_] := N[Exp[(-x)], $MachinePrecision]

Derivation

1. Initial program 7.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 7.5

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

4. Applied rewrites 7.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lft-mult-inverse N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. mul-1-neg N/A

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

   7. associate-*l* N/A

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

   8. mul-1-neg N/A

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

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

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

7. Applied rewrites 7.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-*.f64 N/A

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

9. Applied rewrites 7.3%

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

10. Taylor expanded in x around inf

    \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
11. Step-by-step derivation

    1. neg-mul-1 N/A

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

    2. lower-neg.f64 60.2

       \[\leadsto e^{\color{blue}{-x}} \]

12. Applied rewrites 60.2%

    \[\leadsto e^{\color{blue}{-x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))