expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 43.3%

Time: 18.3s

Alternatives: 8

Speedup: 1.2×

• could not determine a ground truth

  1. x = -1.2957780928449115e+102

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: 6.9% 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: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \frac{\left(\left(e^{x}\right) \bmod \left(t\_0 \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e^{t\_0}}\right)\right)\right)}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (/
    (fmod (exp x) (+ (* t_0 0.6666666666666666) (log (cbrt (exp t_0)))))
    (exp x))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	return fmod(exp(x), ((t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x);
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	return Float64(rem(exp(x), Float64(Float64(t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(t$95$0 * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

   1. add-log-exp 8.3%

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

   2. add-cube-cbrt 42.8%

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

   3. log-prod 42.8%

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

   4. pow2 42.8%

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

6. Applied egg-rr 42.8%

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

   1. unpow2 42.8%

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

   2. log-prod 42.8%

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

   3. pow1/3 42.8%

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

   4. log-pow 42.8%

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

   5. add-log-exp 42.8%

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

   6. pow1/3 42.9%

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

   7. log-pow 42.9%

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

   8. add-log-exp 42.9%

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

8. Applied egg-rr 42.9%

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

   1. distribute-rgt-out 42.9%

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

   2. metadata-eval 42.9%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

10. Simplified 42.9%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

11. Final simplification 42.9%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
12. Add Preprocessing

Alternative 2: 42.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 0.6666666666666666 (log (cbrt E)))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (0.6666666666666666 + log(cbrt(((double) M_E))))) / exp(x);
}

Julia

function code(x)
	return Float64(rem(exp(x), Float64(0.6666666666666666 + log(cbrt(exp(1))))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(0.6666666666666666 + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

   1. add-log-exp 8.3%

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

   2. add-cube-cbrt 42.8%

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

   3. log-prod 42.8%

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

   4. pow2 42.8%

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

6. Applied egg-rr 42.8%

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

   1. unpow2 42.8%

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

   2. log-prod 42.8%

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

   3. pow1/3 42.8%

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

   4. log-pow 42.8%

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

   5. add-log-exp 42.8%

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

   6. pow1/3 42.9%

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

   7. log-pow 42.9%

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

   8. add-log-exp 42.9%

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

8. Applied egg-rr 42.9%

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

   1. distribute-rgt-out 42.9%

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

   2. metadata-eval 42.9%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

10. Simplified 42.9%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

11. Taylor expanded in x around 0 42.4%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e^{1}}\right)\right)}\right)}{e^{x}} \]
12. Step-by-step derivation

    1. exp-1-e 42.4%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

13. Simplified 42.4%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]

14. Final simplification 42.4%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}} \]
15. Add Preprocessing

Alternative 3: 42.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.5e-17)
   (/ (fmod (exp x) 1.0) (exp x))
   (fmod (exp x) (+ 0.6666666666666666 (log (cbrt E))))))

C

double code(double x) {
	double tmp;
	if (x <= -5.5e-17) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = fmod(exp(x), (0.6666666666666666 + log(cbrt(((double) M_E)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.5e-17)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = rem(exp(x), Float64(0.6666666666666666 + log(cbrt(exp(1)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5.5e-17], 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 = N[Exp[x], $MachinePrecision], TMP2 = N[(0.6666666666666666 + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000001e-17

   1. Initial program 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-/r/ 99.7%

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

      3. exp-neg 100.0%

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

      4. remove-double-neg 100.0%

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

   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. Taylor expanded in x around 0 100.0%

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

   if -5.50000000000000001e-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. /-rgt-identity 5.0%

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

      2. associate-/r/ 5.0%

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

      3. exp-neg 5.0%

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

      4. remove-double-neg 5.0%

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

   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. add-log-exp 5.0%

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

      2. add-cube-cbrt 40.7%

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

      3. log-prod 40.7%

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

      4. pow2 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. unpow2 40.7%

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

      2. log-prod 40.7%

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

      3. pow1/3 40.7%

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

      4. log-pow 40.7%

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

      5. add-log-exp 40.7%

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

      6. pow1/3 40.8%

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

      7. log-pow 40.8%

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

      8. add-log-exp 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. distribute-rgt-out 40.8%

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

      2. metadata-eval 40.8%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. Simplified 40.8%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   11. Taylor expanded in x around 0 40.4%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e^{1}}\right)\right)}\right)}{e^{x}} \]
   12. Step-by-step derivation

       1. exp-1-e 40.4%

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

   13. Simplified 40.4%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]

   14. Taylor expanded in x around 0 40.3%

       \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (exp x) (+ 0.6666666666666666 (log (cbrt E)))) (- 1.0 x)))

C

double code(double x) {
	return fmod(exp(x), (0.6666666666666666 + log(cbrt(((double) M_E))))) * (1.0 - x);
}

Julia

function code(x)
	return Float64(rem(exp(x), Float64(0.6666666666666666 + log(cbrt(exp(1))))) * Float64(1.0 - x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(0.6666666666666666 + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

   1. add-log-exp 8.3%

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

   2. add-cube-cbrt 42.8%

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

   3. log-prod 42.8%

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

   4. pow2 42.8%

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

6. Applied egg-rr 42.8%

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

   1. unpow2 42.8%

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

   2. log-prod 42.8%

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

   3. pow1/3 42.8%

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

   4. log-pow 42.8%

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

   5. add-log-exp 42.8%

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

   6. pow1/3 42.9%

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

   7. log-pow 42.9%

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

   8. add-log-exp 42.9%

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

8. Applied egg-rr 42.9%

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

   1. distribute-rgt-out 42.9%

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

   2. metadata-eval 42.9%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

10. Simplified 42.9%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

11. Taylor expanded in x around 0 42.4%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e^{1}}\right)\right)}\right)}{e^{x}} \]
12. Step-by-step derivation

    1. exp-1-e 42.4%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

13. Simplified 42.4%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)}\right)}{e^{x}} \]

14. Taylor expanded in x around 0 41.6%

    \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)} \]
15. Step-by-step derivation

    1. associate-*r* 41.6%

       \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)} + \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \]

    2. neg-mul-1 41.6%

       \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \]

    3. distribute-lft1-in 41.6%

       \[\leadsto \color{blue}{\left(\left(-x\right) + 1\right) \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)} \]

    4. +-commutative 41.6%

       \[\leadsto \color{blue}{\left(1 + \left(-x\right)\right)} \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \]

    5. sub-neg 41.6%

       \[\leadsto \color{blue}{\left(1 - x\right)} \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \]

16. Simplified 41.6%

    \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)} \]

17. Final simplification 41.6%

    \[\leadsto \left(\left(e^{x}\right) \bmod \left(0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right) \cdot \left(1 - x\right) \]
18. Add Preprocessing

Alternative 5: 6.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 1.0 (* (* x x) -0.25))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (1.0 + ((x * x) * -0.25))) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), (1.0d0 + ((x * x) * (-0.25d0)))) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), (1.0 + ((x * x) * -0.25))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), Float64(1.0 + Float64(Float64(x * x) * -0.25))) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

5. Taylor expanded in x around 0 8.2%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
6. Step-by-step derivation

   1. *-commutative 8.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} \]

7. Simplified 8.2%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot -0.25\right)}\right)}{e^{x}} \]
8. Step-by-step derivation

   1. unpow2 8.2%

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

9. Applied egg-rr 8.2%

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

10. Final simplification 8.2%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} \]
11. Add Preprocessing

Alternative 6: 6.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

5. Taylor expanded in x around 0 7.8%

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

6. Final simplification 7.8%

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

Alternative 7: 5.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

5. Taylor expanded in x around 0 7.8%

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

6. Taylor expanded in x around 0 7.0%

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

   1. +-commutative 7.0%

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

   2. *-lft-identity 7.0%

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

   3. associate-*r* 7.0%

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

   4. neg-mul-1 7.0%

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

   5. distribute-rgt-out 7.0%

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

   6. sub-neg 7.0%

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

8. Simplified 7.0%

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

9. Final simplification 7.0%

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

Alternative 8: 5.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. /-rgt-identity 8.3%

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

   2. associate-/r/ 8.3%

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

   3. exp-neg 8.3%

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

   4. remove-double-neg 8.3%

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

3. Simplified 8.3%

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

5. Taylor expanded in x around 0 7.8%

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

6. Taylor expanded in x around 0 6.2%

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

7. Final simplification 6.2%

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

Reproduce

herbie shell --seed 2024048 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))