expfmod (used to be hard to sample)

Percentage Accurate: 7.3% → 61.9%

Time: 20.3s

Alternatives: 6

Speedup: 505.0×

• could not determine a ground truth

  1. x = -3.1242434861414864e+218

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: 7.3% 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: 61.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-310) 1.0 (fmod (+ x 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = fmod((x + 1.0), 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = 1.0d0
    else
        tmp = mod((x + 1.0d0), 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = 1.0
	else:
		tmp = math.fmod((x + 1.0), 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = 1.0;
	else
		tmp = rem(Float64(x + 1.0), 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 13.5%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 13.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 13.5%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 13.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 94.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 94.7%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < x

   1. Initial program 6.3%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 6.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{\left(\sqrt{\cos x}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\color{blue}{\cos x}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right) \]

      4. cos-lowering-cos.f64 5.3%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right) \]

   7. Simplified 5.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 5.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 35.3%

          \[\leadsto \mathsf{fmod.f64}\left(\mathsf{+.f64}\left(1, x\right), 1\right) \]

   13. Simplified 35.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{0 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1e-16) 1.0 (exp (- 0.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-16) {
		tmp = 1.0;
	} else {
		tmp = exp((0.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-16)) then
        tmp = 1.0d0
    else
        tmp = exp((0.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-16) {
		tmp = 1.0;
	} else {
		tmp = Math.exp((0.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-16:
		tmp = 1.0
	else:
		tmp = math.exp((0.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-16)
		tmp = 1.0;
	else
		tmp = exp(Float64(0.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-16)
		tmp = 1.0;
	else
		tmp = exp((0.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-16], 1.0, N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999998e-17

   1. Initial program 99.7%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 50.2%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 50.2%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -9.9999999999999998e-17 < x

   1. Initial program 5.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 5.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 5.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 5.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 5.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 60.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 60.8%

      \[\leadsto e^{\color{blue}{0 - x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.0001) 1.0 (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0001) {
		tmp = 1.0;
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0001d0) then
        tmp = 1.0d0
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.0001:
		tmp = 1.0
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0001)
		tmp = 1.0;
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0001], 1.0, 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 x < 1.00000000000000005e-4

   1. Initial program 11.3%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 11.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 11.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 11.3%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 11.3%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 52.3%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 52.3%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

   12. Simplified 55.1%

       \[\leadsto \color{blue}{1} \]

   if 1.00000000000000005e-4 < x

   1. Initial program 0.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 0.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0

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

      1. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\left(e^{x}\right), \color{blue}{\left(\sqrt{\cos x}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\color{blue}{\cos x}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right) \]

      4. cos-lowering-cos.f64 0.0%

         \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right) \]

   7. Simplified 0.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 0.0%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fmod.f64}\left(\color{blue}{1}, 1\right) \]
   12. Step-by-step derivation

   13. Simplified 100.0%

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

Alternative 4: 47.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 - t\_1}{1 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)} \cdot \left(1 + t\_0 \cdot \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + t\_1} \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ x 1.0))) (t_1 (* x (* x x))))
   (if (<= x 5e+50)
     1.0
     (if (<= x 4e+76)
       (*
        (/ (- 1.0 t_1) (+ 1.0 (* t_0 (* t_0 t_0))))
        (+ 1.0 (* t_0 (+ t_0 -1.0))))
       (* (/ (- 1.0 (* x x)) (+ 1.0 t_1)) (+ 1.0 (* x (+ x -1.0))))))))

C

double code(double x) {
	double t_0 = x * (x + 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= 5e+50) {
		tmp = 1.0;
	} else if (x <= 4e+76) {
		tmp = ((1.0 - t_1) / (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 + (t_0 * (t_0 + -1.0)));
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + t_1)) * (1.0 + (x * (x + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    t_1 = x * (x * x)
    if (x <= 5d+50) then
        tmp = 1.0d0
    else if (x <= 4d+76) then
        tmp = ((1.0d0 - t_1) / (1.0d0 + (t_0 * (t_0 * t_0)))) * (1.0d0 + (t_0 * (t_0 + (-1.0d0))))
    else
        tmp = ((1.0d0 - (x * x)) / (1.0d0 + t_1)) * (1.0d0 + (x * (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x + 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= 5e+50) {
		tmp = 1.0;
	} else if (x <= 4e+76) {
		tmp = ((1.0 - t_1) / (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 + (t_0 * (t_0 + -1.0)));
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + t_1)) * (1.0 + (x * (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x + 1.0)
	t_1 = x * (x * x)
	tmp = 0
	if x <= 5e+50:
		tmp = 1.0
	elif x <= 4e+76:
		tmp = ((1.0 - t_1) / (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 + (t_0 * (t_0 + -1.0)))
	else:
		tmp = ((1.0 - (x * x)) / (1.0 + t_1)) * (1.0 + (x * (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x + 1.0))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 5e+50)
		tmp = 1.0;
	elseif (x <= 4e+76)
		tmp = Float64(Float64(Float64(1.0 - t_1) / Float64(1.0 + Float64(t_0 * Float64(t_0 * t_0)))) * Float64(1.0 + Float64(t_0 * Float64(t_0 + -1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + t_1)) * Float64(1.0 + Float64(x * Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x + 1.0);
	t_1 = x * (x * x);
	tmp = 0.0;
	if (x <= 5e+50)
		tmp = 1.0;
	elseif (x <= 4e+76)
		tmp = ((1.0 - t_1) / (1.0 + (t_0 * (t_0 * t_0)))) * (1.0 + (t_0 * (t_0 + -1.0)));
	else
		tmp = ((1.0 - (x * x)) / (1.0 + t_1)) * (1.0 + (x * (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+50], 1.0, If[LessEqual[x, 4e+76], N[(N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5e50

   1. Initial program 11.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 11.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 11.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 11.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 11.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 53.6%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 53.6%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

   12. Simplified 53.7%

       \[\leadsto \color{blue}{1} \]

   if 5e50 < x < 4.0000000000000002e76

   1. Initial program 0.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 0.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 0.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 0.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{x} \]

       3. --lowering--.f64 2.7%

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   12. Simplified 2.7%

       \[\leadsto \color{blue}{1 - x} \]

   14. Applied egg-rr 100.0%

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

   if 4.0000000000000002e76 < x

   1. Initial program 0.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 0.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 0.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 0.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{x} \]

       3. --lowering--.f64 2.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   12. Simplified 2.0%

       \[\leadsto \color{blue}{1 - x} \]
   13. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + x}} \]

       2. +-commutative N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{x + \color{blue}{1}} \]

       3. flip3-+ N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{\frac{{x}^{3} + {1}^{3}}{\color{blue}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{{x}^{3} + {1}^{3}} \cdot \color{blue}{\left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - x \cdot x}{{x}^{3} + {1}^{3}}\right), \color{blue}{\left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot 1 - x \cdot x\right), \left({x}^{3} + {1}^{3}\right)\right), \left(\color{blue}{x \cdot x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - x \cdot x\right), \left({x}^{3} + {1}^{3}\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot x\right)\right), \left({x}^{3} + {1}^{3}\right)\right), \left(\color{blue}{x} \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{3} + {1}^{3}\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{3} + 1\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left(1 + {x}^{3}\right)\right), \left(x \cdot \color{blue}{x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left({x}^{3}\right)\right)\right), \left(x \cdot \color{blue}{x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 - \color{blue}{x} \cdot 1\right)\right)\right) \]

       17. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 - x\right)\right)\right) \]

       18. associate-+r- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left(x \cdot x + 1\right) - \color{blue}{x}\right)\right) \]

       19. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left(1 + x \cdot x\right) - x\right)\right) \]

       20. associate-+r- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(1 + \color{blue}{\left(x \cdot x - x\right)}\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot x - x\right)}\right)\right) \]

   14. Applied egg-rr 16.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \left(x \cdot \left(x + 1\right)\right) \cdot \left(\left(x \cdot \left(x + 1\right)\right) \cdot \left(x \cdot \left(x + 1\right)\right)\right)} \cdot \left(1 + \left(x \cdot \left(x + 1\right)\right) \cdot \left(x \cdot \left(x + 1\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot \left(x \cdot x\right)} \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.3% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+99}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot \left(x \cdot x\right)} \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+99)
   1.0
   (* (/ (- 1.0 (* x x)) (+ 1.0 (* x (* x x)))) (+ 1.0 (* x (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+99) {
		tmp = 1.0;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + (x * (x * x)))) * (1.0 + (x * (x + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+99) then
        tmp = 1.0d0
    else
        tmp = ((1.0d0 - (x * x)) / (1.0d0 + (x * (x * x)))) * (1.0d0 + (x * (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+99) {
		tmp = 1.0;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + (x * (x * x)))) * (1.0 + (x * (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+99:
		tmp = 1.0
	else:
		tmp = ((1.0 - (x * x)) / (1.0 + (x * (x * x)))) * (1.0 + (x * (x + -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+99)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + Float64(x * Float64(x * x)))) * Float64(1.0 + Float64(x * Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+99)
		tmp = 1.0;
	else
		tmp = ((1.0 - (x * x)) / (1.0 + (x * (x * x)))) * (1.0 + (x * (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+99], 1.0, N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999997e98

   1. Initial program 10.5%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 10.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 10.5%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 10.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 10.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 55.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 55.7%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

   12. Simplified 51.5%

       \[\leadsto \color{blue}{1} \]

   if 9.9999999999999997e98 < x

   1. Initial program 0.0%

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

      5. fmod-lowering-fmod.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

      9. exp-lowering-exp.f64 0.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

   3. Simplified 0.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      11. fmod-lowering-fmod.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 0.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{0 - x}} \]

   10. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto 1 - \color{blue}{x} \]

       3. --lowering--.f64 1.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   12. Simplified 1.9%

       \[\leadsto \color{blue}{1 - x} \]
   13. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + x}} \]

       2. +-commutative N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{x + \color{blue}{1}} \]

       3. flip3-+ N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{\frac{{x}^{3} + {1}^{3}}{\color{blue}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{1 \cdot 1 - x \cdot x}{{x}^{3} + {1}^{3}} \cdot \color{blue}{\left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - x \cdot x}{{x}^{3} + {1}^{3}}\right), \color{blue}{\left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot 1 - x \cdot x\right), \left({x}^{3} + {1}^{3}\right)\right), \left(\color{blue}{x \cdot x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - x \cdot x\right), \left({x}^{3} + {1}^{3}\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot x\right)\right), \left({x}^{3} + {1}^{3}\right)\right), \left(\color{blue}{x} \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{3} + {1}^{3}\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{3} + 1\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left(1 + {x}^{3}\right)\right), \left(x \cdot \color{blue}{x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left({x}^{3}\right)\right)\right), \left(x \cdot \color{blue}{x} + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 - \color{blue}{x} \cdot 1\right)\right)\right) \]

       17. *-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(x \cdot x + \left(1 - x\right)\right)\right) \]

       18. associate-+r- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left(x \cdot x + 1\right) - \color{blue}{x}\right)\right) \]

       19. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\left(1 + x \cdot x\right) - x\right)\right) \]

       20. associate-+r- N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(1 + \color{blue}{\left(x \cdot x - x\right)}\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot x - x\right)}\right)\right) \]

   14. Applied egg-rr 18.5%

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

Alternative 6: 43.4% accurate, 505.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 9.4%

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right), \color{blue}{\left(e^{x}\right)}\right) \]

   5. fmod-lowering-fmod.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right), \left(e^{\color{blue}{x}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right), \left(e^{x}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\cos x\right)\right), \left(e^{x}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \left(e^{x}\right)\right) \]

   9. exp-lowering-exp.f64 9.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{exp.f64}\left(x\right)\right) \]

3. Simplified 9.4%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot -1\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log \left(\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\right)\right) \]

   7. log-div N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\log \left(e^{x}\right) - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

   8. rem-log-exp N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

   10. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

   11. fmod-lowering-fmod.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\left(e^{x}\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

   12. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\sqrt{\cos x}\right)\right)\right)\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \left({\cos x}^{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\cos x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   15. cos-lowering-cos.f64 9.4%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(\mathsf{fmod.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 9.4%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

   3. --lowering--.f64 60.3%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

9. Simplified 60.3%

   \[\leadsto e^{\color{blue}{0 - x}} \]

10. Taylor expanded in x around 0

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

12. Simplified 46.4%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))