expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 26.6%

Time: 11.6s

Alternatives: 11

Speedup: 3.9×

• could not determine a ground truth

  1. x = -2.2629389083130055e+36

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: 7.0% 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: 26.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;e^{-x} \cdot t\_0 \leq 2:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* (exp (- x)) t_0) 2.0)
     (/ 1.0 (/ (exp x) t_0))
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((exp(-x) * t_0) <= 2.0) {
		tmp = 1.0 / (exp(x) / t_0);
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    if ((exp(-x) * t_0) <= 2.0d0) then
        tmp = 1.0d0 / (exp(x) / t_0)
    else
        tmp = 1.0d0 * mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (math.exp(-x) * t_0) <= 2.0:
		tmp = 1.0 / (math.exp(x) / t_0)
	else:
		tmp = 1.0 * math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * t_0) <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / t_0));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{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]}, If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 26.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;e^{-x} \cdot t\_0 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* (exp (- x)) t_0) 2.0) (/ t_0 (exp x)) (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((exp(-x) * t_0) <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    if ((exp(-x) * t_0) <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = 1.0d0 * mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (math.exp(-x) * t_0) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = 1.0 * math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * t_0) <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{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]}, If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   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 9.0

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

   4. Applied rewrites 9.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 26.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (*
      (fmod
       (exp x)
       (sqrt
        (fma
         (fma
          (fma -0.001388888888888889 (* x x) 0.041666666666666664)
          (* x x)
          -0.5)
         (* x x)
         1.0)))
      t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), sqrt(fma(fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5), (x * x), 1.0))) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5), Float64(x * x), 1.0))) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2}} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)}}\right)\right) \cdot e^{-x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      14. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right)}\right)\right) \cdot e^{-x} \]

      15. lower-*.f64 9.0

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 9.0%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(fma(-0.003298611111111111, Float64(x * x), -0.010416666666666666), Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision] + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 9.0

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

   5. Applied rewrites 9.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.4%

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

Alternative 5: 26.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (*
      (fmod
       (exp x)
       (sqrt (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0)))
      t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), sqrt(fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0))) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0))) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}}\right)\right) \cdot e^{-x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1}}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)}}\right)\right) \cdot e^{-x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right)}\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 8.9

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)}\right)\right) \cdot e^{-x} \]

   5. Applied rewrites 8.9%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right)\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (*
      (fmod
       (exp x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 8.9

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

   5. Applied rewrites 8.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.4%

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

Alternative 7: 26.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* t_0 (fmod (exp x) (sqrt (cos x)))) 2.0)
     (* (fmod (exp x) (fma (* x x) -0.25 1.0)) t_0)
     (* 1.0 (fmod 1.0 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((t_0 * fmod(exp(x), sqrt(cos(x)))) <= 2.0) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * t_0;
	} else {
		tmp = 1.0 * fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(t_0 * rem(exp(x), sqrt(cos(x)))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * t_0);
	else
		tmp = Float64(1.0 * rem(1.0, 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $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))) < 2

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 8.8

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

   5. Applied rewrites 8.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.2%

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

4. Final simplification 27.3%

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

Alternative 8: 25.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(Float64(1.0 + x), 1.0) / Float64(1.0 + x))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-exp.f64 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 6.7

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

7. Applied rewrites 6.7%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 5.9

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

10. Applied rewrites 5.9%

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

11. Taylor expanded in x around 0

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

    1. lower-+.f64 25.5

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

13. Applied rewrites 25.5%

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

Alternative 9: 24.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := 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. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

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

8. Applied rewrites 5.3%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 24.2

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

11. Applied rewrites 24.2%

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

12. Taylor expanded in x around 0

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

    1. neg-mul-1 N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 24.6

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

14. Applied rewrites 24.6%

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

15. Final simplification 24.6%

    \[\leadsto \left(1 - x\right) \cdot \left(\left(1 + x\right) \bmod 1\right) \]
16. Add Preprocessing

Alternative 10: 24.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(Float64(1.0 + x), 1.0) * 1.0)
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

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

8. Applied rewrites 5.3%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 24.2

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

11. Applied rewrites 24.2%

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

Alternative 11: 22.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 24.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 23.9%

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

12. Final simplification 23.9%

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

Reproduce

herbie shell --seed 2024241 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))