expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 63.2%

Time: 9.9s

Alternatives: 8

Speedup: 4.1×

• could not determine a ground truth

  1. x = -3.03341171573012e+128

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: 7.1% 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: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(\frac{{\left(\mathsf{fma}\left(-0.5, x, -1\right)\right)}^{-1} \cdot \mathsf{fma}\left(0.015625, {x}^{6}, -1\right)}{\mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fmod 1.0 (pow (fma -0.5 x 1.0) 2.0))
   (if (<= x 50.0)
     (fmod
      (*
       (/
        (* (pow (fma -0.5 x -1.0) -1.0) (fma 0.015625 (pow x 6.0) -1.0))
        (fma -0.25 (hypot (* (* x x) 0.5) x) 1.0))
       x)
      (sqrt (cos x)))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(1.0, pow(fma(-0.5, x, 1.0), 2.0));
	} else if (x <= 50.0) {
		tmp = fmod((((pow(fma(-0.5, x, -1.0), -1.0) * fma(0.015625, pow(x, 6.0), -1.0)) / fma(-0.25, hypot(((x * x) * 0.5), x), 1.0)) * x), sqrt(cos(x)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(1.0, (fma(-0.5, x, 1.0) ^ 2.0));
	elseif (x <= 50.0)
		tmp = rem(Float64(Float64(Float64((fma(-0.5, x, -1.0) ^ -1.0) * fma(0.015625, (x ^ 6.0), -1.0)) / fma(-0.25, hypot(Float64(Float64(x * x) * 0.5), x), 1.0)) * x), sqrt(cos(x)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = 1.0, TMP2 = N[Power[N[(-0.5 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 50.0], N[With[{TMP1 = N[(N[(N[(N[Power[N[(-0.5 * x + -1.0), $MachinePrecision], -1.0], $MachinePrecision] * N[(0.015625 * N[Power[x, 6.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(-0.25 * N[Sqrt[N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.8%

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

   13. Applied rewrites 13.4%

       \[\leadsto \color{blue}{\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)} \]

   if -4.999999999999985e-310 < x < 50

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.4%

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

   13. Applied rewrites 100.0%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left(0.015625, {x}^{6}, -1\right) \cdot {\left(\mathsf{fma}\left(-0.5, x, -1\right)\right)}^{-1}}{\mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

   if 50 < 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

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(\frac{{\left(\mathsf{fma}\left(-0.5, x, -1\right)\right)}^{-1} \cdot \mathsf{fma}\left(0.015625, {x}^{6}, -1\right)}{\mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(0.015625, {x}^{6}, -1\right)}{\mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x, -1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fmod 1.0 (pow (fma -0.5 x 1.0) 2.0))
   (if (<= x 50.0)
     (fmod
      (*
       (/
        (fma 0.015625 (pow x 6.0) -1.0)
        (* (fma -0.25 (hypot (* (* x x) 0.5) x) 1.0) (fma -0.5 x -1.0)))
       x)
      (sqrt (cos x)))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(1.0, pow(fma(-0.5, x, 1.0), 2.0));
	} else if (x <= 50.0) {
		tmp = fmod(((fma(0.015625, pow(x, 6.0), -1.0) / (fma(-0.25, hypot(((x * x) * 0.5), x), 1.0) * fma(-0.5, x, -1.0))) * x), sqrt(cos(x)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(1.0, (fma(-0.5, x, 1.0) ^ 2.0));
	elseif (x <= 50.0)
		tmp = rem(Float64(Float64(fma(0.015625, (x ^ 6.0), -1.0) / Float64(fma(-0.25, hypot(Float64(Float64(x * x) * 0.5), x), 1.0) * fma(-0.5, x, -1.0))) * x), sqrt(cos(x)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = 1.0, TMP2 = N[Power[N[(-0.5 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 50.0], N[With[{TMP1 = N[(N[(N[(0.015625 * N[Power[x, 6.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(-0.25 * N[Sqrt[N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.8%

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

   13. Applied rewrites 13.4%

       \[\leadsto \color{blue}{\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)} \]

   if -4.999999999999985e-310 < x < 50

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.4%

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

   13. Applied rewrites 100.0%

       \[\leadsto \left(\left(\frac{\mathsf{fma}\left(0.015625, {x}^{6}, -1\right)}{\mathsf{fma}\left(-0.5, x, -1\right) \cdot \mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

   if 50 < 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

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(0.015625, {x}^{6}, -1\right)}{\mathsf{fma}\left(-0.25, \mathsf{hypot}\left(\left(x \cdot x\right) \cdot 0.5, x\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x, -1\right)} \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 50:\\ \;\;\;\;\left(\left(1 \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (fmod 1.0 (pow (fma -0.5 x 1.0) 2.0))
   (if (<= x 50.0) (fmod (* 1.0 x) (sqrt (cos x))) (fmod 1.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(1.0, pow(fma(-0.5, x, 1.0), 2.0));
	} else if (x <= 50.0) {
		tmp = fmod((1.0 * x), sqrt(cos(x)));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = rem(1.0, (fma(-0.5, x, 1.0) ^ 2.0));
	elseif (x <= 50.0)
		tmp = rem(Float64(1.0 * x), sqrt(cos(x)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[With[{TMP1 = 1.0, TMP2 = N[Power[N[(-0.5 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[x, 50.0], N[With[{TMP1 = N[(1.0 * x), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 11.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.8%

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

   13. Applied rewrites 13.4%

       \[\leadsto \color{blue}{\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)} \]

   if -4.999999999999985e-310 < x < 50

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 99.4%

       \[\leadsto \left(\left(1 \cdot x\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

   if 50 < 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

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \left(1 \bmod 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 26.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50:\\ \;\;\;\;\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50.0) (fmod 1.0 (pow (fma -0.5 x 1.0) 2.0)) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 50.0) {
		tmp = fmod(1.0, pow(fma(-0.5, x, 1.0), 2.0));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 50.0)
		tmp = rem(1.0, (fma(-0.5, x, 1.0) ^ 2.0));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 50.0], N[With[{TMP1 = 1.0, TMP2 = N[Power[N[(-0.5 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 6.2

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

   5. Applied rewrites 6.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 6.9%

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

   13. Applied rewrites 10.4%

       \[\leadsto \color{blue}{\left(1 \bmod \left({\left(\mathsf{fma}\left(-0.5, x, 1\right)\right)}^{2}\right)\right)} \]

   if 50 < 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

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 5: 24.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.9)
   (fmod 1.0 (fma (* x x) -0.25 1.0))
   (* (- 1.0 x) (fmod (+ 1.0 x) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.9) {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0));
	} else {
		tmp = (1.0 - x) * fmod((1.0 + x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.9)
		tmp = rem(1.0, fma(Float64(x * x), -0.25, 1.0));
	else
		tmp = Float64(Float64(1.0 - x) * rem(Float64(1.0 + x), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.9], N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.900000000000000022

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.9

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

   if -0.900000000000000022 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 5.7

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

   8. Applied rewrites 5.7%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 24.8

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

   11. Applied rewrites 24.8%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 24.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;\left(1 \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - -1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.82) (fmod 1.0 (fma (* x x) -0.25 1.0)) (fmod (- x -1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -0.82) {
		tmp = fmod(1.0, fma((x * x), -0.25, 1.0));
	} else {
		tmp = fmod((x - -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.82)
		tmp = rem(1.0, fma(Float64(x * x), -0.25, 1.0));
	else
		tmp = rem(Float64(x - -1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.82], N[With[{TMP1 = 1.0, TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = N[(x - -1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.819999999999999951

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.9

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

   if -0.819999999999999951 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-cos.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 24.1%

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

Alternative 7: 23.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 23.7%

    \[\leadsto \left(\left(x - -1\right) \bmod 1\right) \]
12. Add Preprocessing

Alternative 8: 22.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 22.7%

    \[\leadsto \left(1 \bmod 1\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))