expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 62.3%

Time: 15.2s

Alternatives: 7

Speedup: 5.0×

• could not determine a ground truth

  1. x = -6.029758719952058e+26

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (/ (fmod (exp x) (* 3.0 (log (cbrt E)))) (exp x))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(((double) M_E))))) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(3.0 * log(cbrt(exp(1))))) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.3%

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

      1. /-rgt-identity 7.3%

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

      2. associate-/r/ 7.3%

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

      3. exp-neg 7.3%

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

      4. remove-double-neg 7.3%

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

   3. Simplified 7.3%

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

      1. add-log-exp 7.3%

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

      2. add-cube-cbrt 60.3%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right)}\right)}{e^{x}} \]

      3. log-prod 60.3%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

   6. Applied egg-rr 60.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   7. Step-by-step derivation

      1. log-pow 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\color{blue}{1}}}}\right)\right)\right)}{e^{x}} \]

      5. metadata-eval 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\color{blue}{1}}}\right)\right)\right)}{e^{x}} \]

      6. exp-1-e 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   8. Simplified 60.1%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 95.8%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 95.8%

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

      3. metadata-eval 95.8%

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

   8. Simplified 95.8%

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

   9. Taylor expanded in x around 0 96.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (fmod (exp x) (* 3.0 (log (cbrt E))))
   (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(((double) M_E)))));
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = rem(exp(x), Float64(3.0 * log(cbrt(exp(1)))));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0 6.2%

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

      1. add-log-exp 7.3%

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

      2. add-cube-cbrt 60.3%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right)}\right)}{e^{x}} \]

      3. log-prod 60.3%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

   5. Applied egg-rr 59.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. log-pow 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\color{blue}{1}}}}\right)\right)\right)}{e^{x}} \]

      5. metadata-eval 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\color{blue}{1}}}\right)\right)\right)}{e^{x}} \]

      6. exp-1-e 60.1%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   7. Simplified 59.2%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)}\right) \]

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 95.8%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 95.8%

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

      3. metadata-eval 95.8%

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

   8. Simplified 95.8%

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

   9. Taylor expanded in x around 0 96.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 26.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0005) (exp (- (log (fmod (exp x) 1.0)) x)) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = exp((log(fmod(exp(x), 1.0)) - x));
	} 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 <= 0.0005d0) then
        tmp = exp((log(mod(exp(x), 1.0d0)) - x))
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.0005:
		tmp = math.exp((math.log(math.fmod(math.exp(x), 1.0)) - x))
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = exp(Float64(log(rem(exp(x), 1.0)) - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0005], N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 7.3%

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

      1. /-rgt-identity 7.3%

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

      2. associate-/r/ 7.2%

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

      3. exp-neg 7.3%

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

      4. remove-double-neg 7.3%

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

   3. Simplified 7.3%

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

      1. add-log-exp 7.2%

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

      2. add-cube-cbrt 59.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right)}\right)}{e^{x}} \]

      3. log-prod 59.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

   6. Applied egg-rr 59.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   7. Step-by-step derivation

      1. log-pow 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\color{blue}{1}}}}\right)\right)\right)}{e^{x}} \]

      5. metadata-eval 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\color{blue}{1}}}\right)\right)\right)}{e^{x}} \]

      6. exp-1-e 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   8. Simplified 59.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]
   9. Step-by-step derivation

      1. add-log-exp 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\log \left(e^{3 \cdot \log \left(\sqrt[3]{e}\right)}\right)}\right)}{e^{x}} \]

      2. *-commutative 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \left(e^{\color{blue}{\log \left(\sqrt[3]{e}\right) \cdot 3}}\right)\right)}{e^{x}} \]

      3. exp-to-pow 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left({\left(\sqrt[3]{e}\right)}^{3}\right)}\right)}{e^{x}} \]

      4. pow3 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left(\left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \sqrt[3]{e}\right)}\right)}{e^{x}} \]

      5. add-cube-cbrt 7.0%

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

      6. log-E 7.0%

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

      7. add-exp-log 7.0%

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

      8. div-exp 7.1%

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

   10. Applied egg-rr 7.1%

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

   if 5.0000000000000001e-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. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0005) (/ (fmod (exp x) 1.0) (exp x)) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0005d0) then
        tmp = mod(exp(x), 1.0d0) / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.0005:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0005], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 7.3%

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

      1. /-rgt-identity 7.3%

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

      2. associate-/r/ 7.2%

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

      3. exp-neg 7.3%

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

      4. remove-double-neg 7.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in x around 0 7.0%

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

   if 5.0000000000000001e-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. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 25.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0005) (* (fmod (exp x) 1.0) (- 1.0 x)) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} 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 <= 0.0005d0) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.0005:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0005], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 7.3%

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

      1. /-rgt-identity 7.3%

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

      2. associate-/r/ 7.2%

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

      3. exp-neg 7.3%

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

      4. remove-double-neg 7.3%

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

   3. Simplified 7.3%

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

      1. add-log-exp 7.2%

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

      2. add-cube-cbrt 59.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right)}\right)}{e^{x}} \]

      3. log-prod 59.7%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

   6. Applied egg-rr 59.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]
   7. Step-by-step derivation

      1. log-pow 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      2. distribute-lft1-in 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)}\right)}{e^{x}} \]

      3. metadata-eval 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos 0}}}\right)\right)\right)}{e^{x}} \]

      4. cos-0 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\color{blue}{1}}}}\right)\right)\right)}{e^{x}} \]

      5. metadata-eval 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\color{blue}{1}}}\right)\right)\right)}{e^{x}} \]

      6. exp-1-e 59.5%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   8. Simplified 59.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]

   9. Taylor expanded in x around 0 6.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 6.9%

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} + \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right) \]

       2. neg-mul-1 6.9%

          \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right) \]

       3. distribute-lft1-in 6.9%

          \[\leadsto \color{blue}{\left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left({e}^{0.3333333333333333}\right)\right)\right)} \]

       4. log-pow 6.9%

          \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log e\right)}\right)\right) \]

       5. log-E 6.9%

          \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{1}\right)\right)\right) \]

       6. metadata-eval 6.9%

          \[\leadsto \left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(3 \cdot \color{blue}{0.3333333333333333}\right)\right) \]

       7. metadata-eval 6.9%

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

       8. distribute-rgt1-in 6.9%

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

       9. *-lft-identity 6.9%

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

       10. distribute-rgt-out 6.9%

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

       11. unsub-neg 6.9%

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

   11. Simplified 6.9%

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

   if 5.0000000000000001e-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. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 25.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0005) (fmod (exp x) 1.0) (fmod 1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = fmod(exp(x), 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 <= 0.0005d0) then
        tmp = mod(exp(x), 1.0d0)
    else
        tmp = mod(1.0d0, 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.0005:
		tmp = math.fmod(math.exp(x), 1.0)
	else:
		tmp = math.fmod(1.0, 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = rem(exp(x), 1.0);
	else
		tmp = rem(1.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0005], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0 6.1%

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

   4. Taylor expanded in x around 0 6.2%

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

   if 5.0000000000000001e-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. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   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. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

      5. add-log-exp 0.0%

         \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
   7. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

      2. cos-0 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 23.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

   1. /-rgt-identity 5.9%

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

   2. associate-/r/ 5.9%

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

   3. exp-neg 5.9%

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

   4. remove-double-neg 5.9%

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

3. Simplified 5.9%

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

   1. add-log-exp 5.9%

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

   2. *-un-lft-identity 5.9%

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

   3. log-prod 5.9%

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

   4. metadata-eval 5.9%

      \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}} \]

   5. add-log-exp 5.9%

      \[\leadsto \frac{0 + \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

6. Applied egg-rr 21.2%

   \[\leadsto \frac{\color{blue}{0 + \left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]
7. Step-by-step derivation

   1. +-lft-identity 21.2%

      \[\leadsto \frac{\color{blue}{\left(1 \bmod \left(\sqrt{\cos 0}\right)\right)}}{e^{x}} \]

   2. cos-0 21.2%

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

   3. metadata-eval 21.2%

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

8. Simplified 21.2%

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

9. Taylor expanded in x around 0 21.2%

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

10. Final simplification 21.2%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))