expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 63.3%

Time: 25.1s

Alternatives: 6

Speedup: 4.9×

• could not determine a ground truth

  1. x = -1.9843459328788718e+48

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: 6.8% 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: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310)
   (/
    (fmod
     (+ 1.0 (* x (+ 1.0 (* x 0.5))))
     (sqrt (+ (log (pow (cbrt E) 2.0)) (log (cbrt E)))))
    (exp x))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = fmod((1.0 + (x * (1.0 + (x * 0.5)))), sqrt((log(pow(cbrt(((double) M_E)), 2.0)) + log(cbrt(((double) M_E)))))) / exp(x);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(rem(Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))), sqrt(Float64(log((cbrt(exp(1)) ^ 2.0)) + log(cbrt(exp(1)))))) / exp(x));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[(N[With[{TMP1 = N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[N[Power[E, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 5.7%

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

      1. /-rgt-identity 5.7%

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

      2. associate-/r/ 5.7%

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

      3. exp-neg 5.7%

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

      4. remove-double-neg 5.7%

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

   3. Simplified 5.7%

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

      1. add-log-exp 5.7%

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

      2. add-cube-cbrt 98.2%

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

      3. log-prod 98.2%

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

      4. pow2 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in x around 0 98.2%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \color{blue}{\left(\sqrt[3]{e^{1}}\right)}}\right)\right)}{e^{x}} \]
   8. Step-by-step derivation

      1. exp-1-e 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in x around 0 98.2%

       \[\leadsto \frac{\left(\color{blue}{\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right)} \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]

   12. Simplified 98.2%

       \[\leadsto \frac{\left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right)} \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]

   13. Taylor expanded in x around 0 98.2%

       \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\log \left({\color{blue}{\left(\sqrt[3]{e^{1}}\right)}}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]
   14. Step-by-step derivation

       1. exp-1-e 98.2%

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

   15. Simplified 98.2%

       \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\log \left({\color{blue}{\left(\sqrt[3]{e}\right)}}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]

   if -9.999999999999969e-311 < x

   1. Initial program 4.1%

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

      1. /-rgt-identity 4.1%

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

      2. associate-/r/ 4.1%

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

      3. exp-neg 4.1%

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

      4. remove-double-neg 4.1%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in x around 0 4.0%

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

   6. Taylor expanded in x around 0 38.8%

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

      1. +-commutative 38.8%

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

   8. Simplified 38.8%

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

Alternative 2: 30.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. /-rgt-identity 4.8%

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

   2. associate-/r/ 4.8%

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

   3. exp-neg 4.8%

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

   4. remove-double-neg 4.8%

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

3. Simplified 4.8%

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

5. Taylor expanded in x around 0 24.6%

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

   1. +-commutative 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in x around inf 33.1%

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

Alternative 3: 31.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(\sqrt{\cos x}\right)\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-303)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (sqrt (cos x))) (+ x 1.0))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-303) {
		tmp = fmod((x * (1.0 + (1.0 / x))), sqrt(cos(x))) / (x + 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-303)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), sqrt(cos(x))) / (x + 1.0d0)
    else
        tmp = mod((x + 1.0d0), 1.0d0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-303:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), math.sqrt(math.cos(x))) / (x + 1.0)
	else:
		tmp = math.fmod((x + 1.0), 1.0) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-303)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), sqrt(cos(x))) / Float64(x + 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-303], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999931e-304

   1. Initial program 5.7%

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

      1. /-rgt-identity 5.7%

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

      2. associate-/r/ 5.7%

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

      3. exp-neg 5.8%

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

      4. remove-double-neg 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in x around 0 5.5%

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

      1. +-commutative 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in x around 0 6.0%

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

      1. +-commutative 5.5%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around inf 26.3%

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

   if -9.99999999999999931e-304 < x

   1. Initial program 4.1%

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

      1. /-rgt-identity 4.1%

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

      2. associate-/r/ 4.1%

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

      3. exp-neg 4.1%

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

      4. remove-double-neg 4.1%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in x around 0 4.0%

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

   6. Taylor expanded in x around 0 38.6%

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

      1. +-commutative 38.6%

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

   8. Simplified 38.6%

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

Alternative 4: 29.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. /-rgt-identity 4.8%

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

   2. associate-/r/ 4.8%

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

   3. exp-neg 4.8%

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

   4. remove-double-neg 4.8%

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

3. Simplified 4.8%

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

5. Taylor expanded in x around 0 4.7%

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

6. Taylor expanded in x around 0 4.2%

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

7. Taylor expanded in x around 0 23.3%

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

   1. +-commutative 24.6%

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

9. Simplified 23.3%

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

10. Taylor expanded in x around inf 31.8%

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

Alternative 5: 25.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. /-rgt-identity 4.8%

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

   2. associate-/r/ 4.8%

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

   3. exp-neg 4.8%

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

   4. remove-double-neg 4.8%

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

3. Simplified 4.8%

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

5. Taylor expanded in x around 0 24.6%

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

   1. +-commutative 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in x around 0 5.9%

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

   1. +-commutative 24.6%

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

10. Simplified 5.9%

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

11. Taylor expanded in x around 0 24.0%

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

Alternative 6: 24.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. /-rgt-identity 4.8%

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

   2. associate-/r/ 4.8%

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

   3. exp-neg 4.8%

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

   4. remove-double-neg 4.8%

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

3. Simplified 4.8%

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

5. Taylor expanded in x around 0 4.7%

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

6. Taylor expanded in x around 0 4.2%

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

7. Taylor expanded in x around 0 23.3%

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

   1. +-commutative 24.6%

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

9. Simplified 23.3%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))