expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 26.5%

Time: 15.6s

Alternatives: 16

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.3708373074200754e+283

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 26.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left({\left(0.5 + 0.5 \cdot \cos \left(x \cdot 2\right)\right)}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* t_0 (fmod (exp x) (pow (+ 0.5 (* 0.5 (cos (* x 2.0)))) 0.25)))
     (fmod 1.0 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = t_0 * fmod(exp(x), pow((0.5 + (0.5 * cos((x * 2.0)))), 0.25));
	} else {
		tmp = 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 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = t_0 * mod(exp(x), ((0.5d0 + (0.5d0 * cos((x * 2.0d0)))) ** 0.25d0))
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = t_0 * math.fmod(math.exp(x), math.pow((0.5 + (0.5 * math.cos((x * 2.0)))), 0.25))
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), (Float64(0.5 + Float64(0.5 * cos(Float64(x * 2.0)))) ^ 0.25)));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[(0.5 + N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

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

      1. lift-cos.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. sqr-cos-a N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 10.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left({\left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}^{\color{blue}{0.25}}\right)\right) \cdot e^{-x} \]

   4. Applied egg-rr 10.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}^{0.25}\right)}\right) \cdot e^{-x} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 100.0%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left({\left(0.5 + 0.5 \cdot \cos \left(x \cdot 2\right)\right)}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 26.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((t_0 * exp(-x)) <= 2.0) {
		tmp = 1.0 / (exp(x) / t_0);
	} else {
		tmp = 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 ((t_0 * exp(-x)) <= 2.0d0) then
        tmp = 1.0d0 / (exp(x) / t_0)
    else
        tmp = 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 (t_0 * math.exp(-x)) <= 2.0:
		tmp = 1.0 / (math.exp(x) / t_0)
	else:
		tmp = 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(t_0 * exp(Float64(-x))) <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / t_0));
	else
		tmp = 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[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fmod.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. clear-num 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 N/A

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

      10. lower-/.f64 10.4

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

   4. Applied egg-rr 10.4%

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

   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

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 100.0%

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

Alternative 3: 26.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((t_0 * exp(-x)) <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = 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 ((t_0 * exp(-x)) <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = 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 (t_0 * math.exp(-x)) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = 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(t_0 * exp(Float64(-x))) <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = 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[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fmod.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 10.4

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

   4. Applied egg-rr 10.4%

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

   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

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 100.0%

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

Alternative 4: 26.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x))) * exp(-x);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 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))) * exp(-x)
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = 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))) * math.exp(-x)
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 2.0], t$95$0, 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 (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 10.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-fmod.f64 N/A

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

      2. lower-exp.f64 0.0

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 100.0%

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

Alternative 5: 26.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.003298611111111111 + \frac{-0.010416666666666666 + \frac{-0.25}{x \cdot x}}{x \cdot x}\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -1.25e-81)
     (*
      t_0
      (fmod
       (exp x)
       (fma
        (* x x)
        (*
         (* (* x x) (* x x))
         (+
          -0.003298611111111111
          (/ (+ -0.010416666666666666 (/ -0.25 (* x x))) (* x x))))
        1.0)))
     (* t_0 (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -1.25e-81) {
		tmp = t_0 * fmod(exp(x), fma((x * x), (((x * x) * (x * x)) * (-0.003298611111111111 + ((-0.010416666666666666 + (-0.25 / (x * x))) / (x * x)))), 1.0));
	} else {
		tmp = t_0 * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -1.25e-81)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(-0.003298611111111111 + Float64(Float64(-0.010416666666666666 + Float64(-0.25 / Float64(x * x))) / Float64(x * x)))), 1.0)));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -1.25e-81], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.003298611111111111 + N[(N[(-0.010416666666666666 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24999999999999995e-81

   1. Initial program 23.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 23.9

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

   5. Simplified 23.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   8. Simplified 31.8%

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

   if -1.24999999999999995e-81 < x

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 4.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 28.4

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

   8. Simplified 28.4%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.003298611111111111 + \frac{-0.010416666666666666 + \frac{-0.25}{x \cdot x}}{x \cdot x}\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (/
    1.0
    (/
     (exp x)
     (fmod
      (exp x)
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) -0.003298611111111111 -0.010416666666666666)
         -0.25))
       1.0))))
   (* (exp (- x)) (fmod (+ x 1.0) 1.0))))

C

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / rem(exp(x), fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666), -0.25)), 1.0))));
	else
		tmp = Float64(exp(Float64(-x)) * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 9.8

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

   5. Simplified 9.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fmod.f64 N/A

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

      9. exp-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. un-div-inv N/A

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

   7. Applied egg-rr 9.8%

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

   if 2 < x

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 28.6%

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

Alternative 7: 26.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666)), -0.25), 1.0)));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 9.8

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

   5. Simplified 9.8%

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

   if 2 < x

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 28.6%

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

Alternative 8: 26.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), sqrt(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -0.5), 1.0))));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 2.0], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 9.7

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

   5. Simplified 9.7%

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

   if 2 < x

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 28.5%

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

Alternative 9: 26.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;t\_0 \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x 1.15)
     (*
      t_0
      (fmod
       (exp x)
       (fma (* x x) (fma (* x x) -0.010416666666666666 -0.25) 1.0)))
     (* t_0 (fmod (+ x 1.0) 1.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 1.15) {
		tmp = t_0 * fmod(exp(x), fma((x * x), fma((x * x), -0.010416666666666666, -0.25), 1.0));
	} else {
		tmp = t_0 * fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 1.15)
		tmp = Float64(t_0 * rem(exp(x), fma(Float64(x * x), fma(Float64(x * x), -0.010416666666666666, -0.25), 1.0)));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 1.15], N[(t$95$0 * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 9.7

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

   5. Simplified 9.7%

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

   if 1.1499999999999999 < x

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.010416666666666666, -0.25\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(t_0 * rem(exp(x), fma(x, Float64(x * -0.25), 1.0)));
	else
		tmp = Float64(t_0 * rem(Float64(x + 1.0), 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 9.5

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

   5. Simplified 9.5%

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

   if 2 < x

   1. Initial program 1.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 0.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 28.3%

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

Alternative 11: 25.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 26.6

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

8. Simplified 26.6%

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

   1. lift-+.f64 N/A

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

   2. lift-fmod.f64 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 26.6

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

10. Applied egg-rr 26.6%

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

11. Final simplification 26.6%

    \[\leadsto \frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}} \]
12. Add Preprocessing

Alternative 12: 25.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 26.6

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

8. Simplified 26.6%

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

9. Final simplification 26.6%

   \[\leadsto e^{-x} \cdot \left(\left(x + 1\right) \bmod 1\right) \]
10. Add Preprocessing

Alternative 13: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 26.6

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

8. Simplified 26.6%

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

   1. lift-+.f64 N/A

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

   2. lift-fmod.f64 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 26.6

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

10. Applied egg-rr 26.6%

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

11. Taylor expanded in x around 0

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

    1. lower-+.f64 25.9

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

13. Simplified 25.9%

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

14. Final simplification 25.9%

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

Alternative 14: 24.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 26.6

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

8. Simplified 26.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(1 + x\right) \bmod 1\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
10. 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 25.3

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

11. Simplified 25.3%

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

12. Final simplification 25.3%

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

Alternative 15: 24.3% 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 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 26.6

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

8. Simplified 26.6%

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

9. Taylor expanded in x around 0

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

    1. lower-fmod.f64 N/A

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

    2. lower-+.f64 24.9

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

11. Simplified 24.9%

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

12. Final simplification 24.9%

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

Alternative 16: 23.2% 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 8.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 7.0%

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

6. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 5.3

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

8. Simplified 5.3%

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

9. Taylor expanded in x around 0

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

11. Simplified 23.8%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))