expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 63.3%

Time: 20.4s

Alternatives: 6

Speedup: 5.0×

• 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

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

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) (sqrt (cos x))) (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), sqrt(cos(x))) / 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), sqrt(cos(x))) / 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 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, 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 38.8%

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

      1. +-commutative 38.8%

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

   7. Simplified 38.8%

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

Alternative 2: 28.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \cos x \cdot 0.6666666666666666\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{{\left(\sqrt[3]{-x}\right)}^{3}}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 200.0)
   (/
    (fmod
     (+ 1.0 (* x (+ 1.0 (* x 0.5))))
     (sqrt
      (fma
       (sqrt 0.3333333333333333)
       (sqrt 0.3333333333333333)
       (* (cos x) 0.6666666666666666))))
    (exp x))
   (fmod (exp (pow (cbrt (- x)) 3.0)) (sqrt (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 200.0) {
		tmp = fmod((1.0 + (x * (1.0 + (x * 0.5)))), sqrt(fma(sqrt(0.3333333333333333), sqrt(0.3333333333333333), (cos(x) * 0.6666666666666666)))) / exp(x);
	} else {
		tmp = fmod(exp(pow(cbrt(-x), 3.0)), sqrt(cos(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 200.0)
		tmp = Float64(rem(Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))), sqrt(fma(sqrt(0.3333333333333333), sqrt(0.3333333333333333), Float64(cos(x) * 0.6666666666666666)))) / exp(x));
	else
		tmp = rem(exp((cbrt(Float64(-x)) ^ 3.0)), sqrt(cos(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 200.0], 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[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[0.3333333333333333], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[With[{TMP1 = N[Exp[N[Power[N[Power[(-x), 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 200

   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 6.0%

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

      4. remove-double-neg 6.0%

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

   3. Simplified 6.0%

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

      1. add-log-exp 6.0%

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

      2. add-cube-cbrt 55.2%

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

      3. log-prod 55.2%

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

      4. pow2 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in x around 0 55.2%

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

      1. exp-1-e 55.2%

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

   9. Simplified 55.2%

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

   10. Taylor expanded in x around 0 55.2%

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

       1. *-commutative 55.2%

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

   12. Simplified 55.2%

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

       1. +-commutative 55.2%

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

       2. add-sqr-sqrt 55.2%

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

       3. fma-define 55.2%

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

       4. pow1/3 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{\log \color{blue}{\left({e}^{0.3333333333333333}\right)}}, \sqrt{\log \left(\sqrt[3]{e}\right)}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       5. log-pow 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{\color{blue}{0.3333333333333333 \cdot \log e}}, \sqrt{\log \left(\sqrt[3]{e}\right)}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       6. log-E 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333 \cdot \color{blue}{1}}, \sqrt{\log \left(\sqrt[3]{e}\right)}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       7. metadata-eval 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{\color{blue}{0.3333333333333333}}, \sqrt{\log \left(\sqrt[3]{e}\right)}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       8. pow1/3 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{\log \color{blue}{\left({e}^{0.3333333333333333}\right)}}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       9. log-pow 55.2%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{\color{blue}{0.3333333333333333 \cdot \log e}}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       10. log-E 55.2%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333 \cdot \color{blue}{1}}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       11. metadata-eval 55.2%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{\color{blue}{0.3333333333333333}}, \log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       12. pow1/3 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \log \left({\color{blue}{\left({\left(e^{\cos x}\right)}^{0.3333333333333333}\right)}}^{2}\right)\right)}\right)\right)}{e^{x}} \]

       13. pow-pow 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \log \color{blue}{\left({\left(e^{\cos x}\right)}^{\left(0.3333333333333333 \cdot 2\right)}\right)}\right)}\right)\right)}{e^{x}} \]

       14. metadata-eval 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \log \left({\left(e^{\cos x}\right)}^{\color{blue}{0.6666666666666666}}\right)\right)}\right)\right)}{e^{x}} \]

       15. exp-prod 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \log \color{blue}{\left(e^{\cos x \cdot 0.6666666666666666}\right)}\right)}\right)\right)}{e^{x}} \]

       16. *-commutative 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \log \left(e^{\color{blue}{0.6666666666666666 \cdot \cos x}}\right)\right)}\right)\right)}{e^{x}} \]

       17. add-log-exp 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \color{blue}{0.6666666666666666 \cdot \cos x}\right)}\right)\right)}{e^{x}} \]

       18. *-commutative 9.5%

           \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \color{blue}{\cos x \cdot 0.6666666666666666}\right)}\right)\right)}{e^{x}} \]

   14. Applied egg-rr 9.5%

       \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{0.3333333333333333}, \sqrt{0.3333333333333333}, \cos x \cdot 0.6666666666666666\right)}}\right)\right)}{e^{x}} \]

   if 200 < x

   1. Initial program 0.3%

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

   3. Taylor expanded in x around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 98.6%

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

      6. neg-mul-1 98.6%

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

      7. exp-prod 98.6%

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

      8. add-log-exp 98.6%

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

      9. add-sqr-sqrt 98.6%

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

      10. log-prod 98.6%

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

      11. unpow-prod-up 98.6%

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

      12. pow1/2 98.6%

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

      13. log-pow 98.6%

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

      14. add-log-exp 98.6%

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

      15. pow1/2 98.6%

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

      16. log-pow 98.6%

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

      17. add-log-exp 98.6%

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

   5. Applied egg-rr 98.6%

      \[\leadsto \left(\color{blue}{\left({\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)} \cdot {\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \]
   6. Step-by-step derivation

      1. pow-sqr 98.6%

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

      2. associate-*r* 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-lft-identity 98.6%

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

      5. exp-prod 98.6%

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

      6. neg-mul-1 98.6%

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

   7. Simplified 98.6%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

   9. Applied egg-rr 98.6%

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

Alternative 3: 25.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[(x + 1.0), $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%

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

7. Simplified 24.6%

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

Alternative 4: 25.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left({x}^{2} \cdot \left(\frac{1}{{x}^{2}} - 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{-x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.56e-162)
   (fmod 1.0 (* (pow x 2.0) (- (/ 1.0 (pow x 2.0)) 0.25)))
   (fmod (exp (- x)) (+ 1.0 (* (pow x 2.0) -0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.56e-162) {
		tmp = fmod(1.0, (pow(x, 2.0) * ((1.0 / pow(x, 2.0)) - 0.25)));
	} else {
		tmp = fmod(exp(-x), (1.0 + (pow(x, 2.0) * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.56d-162)) then
        tmp = mod(1.0d0, ((x ** 2.0d0) * ((1.0d0 / (x ** 2.0d0)) - 0.25d0)))
    else
        tmp = mod(exp(-x), (1.0d0 + ((x ** 2.0d0) * (-0.25d0))))
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1.56e-162:
		tmp = math.fmod(1.0, (math.pow(x, 2.0) * ((1.0 / math.pow(x, 2.0)) - 0.25)))
	else:
		tmp = math.fmod(math.exp(-x), (1.0 + (math.pow(x, 2.0) * -0.25)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.56e-162)
		tmp = rem(1.0, Float64((x ^ 2.0) * Float64(Float64(1.0 / (x ^ 2.0)) - 0.25)));
	else
		tmp = rem(exp(Float64(-x)), Float64(1.0 + Float64((x ^ 2.0) * -0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.56e-162], N[With[{TMP1 = 1.0, TMP2 = N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[With[{TMP1 = N[Exp[(-x)], $MachinePrecision], TMP2 = N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5600000000000001e-162

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 5.9%

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

   4. Taylor expanded in x around 0 4.0%

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

   5. Taylor expanded in x around 0 7.0%

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

   6. Taylor expanded in x around inf 11.4%

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

   if -1.5600000000000001e-162 < x

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0 3.7%

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

      1. add-sqr-sqrt 2.9%

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

      2. sqrt-unprod 3.7%

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

      3. sqr-neg 3.7%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 28.9%

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

      6. neg-mul-1 28.9%

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

      7. exp-prod 28.9%

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

      8. add-log-exp 28.9%

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

      9. add-sqr-sqrt 28.9%

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

      10. log-prod 28.9%

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

      11. unpow-prod-up 28.9%

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

      12. pow1/2 28.9%

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

      13. log-pow 28.9%

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

      14. add-log-exp 28.9%

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

      15. pow1/2 28.9%

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

      16. log-pow 28.9%

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

      17. add-log-exp 28.9%

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

   5. Applied egg-rr 28.9%

      \[\leadsto \left(\color{blue}{\left({\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)} \cdot {\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \]
   6. Step-by-step derivation

      1. pow-sqr 28.9%

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

      2. associate-*r* 28.9%

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

      3. metadata-eval 28.9%

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

      4. *-lft-identity 28.9%

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

      5. exp-prod 28.9%

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

      6. neg-mul-1 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in x around 0 28.9%

      \[\leadsto \left(\left(e^{-x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-162}:\\ \;\;\;\;\left(1 \bmod \left({x}^{2} \cdot \left(\frac{1}{{x}^{2}} - 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{-x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 23.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{-x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fmod (exp (- x)) (+ 1.0 (* (pow x 2.0) -0.25))))

C

double code(double x) {
	return fmod(exp(-x), (1.0 + (pow(x, 2.0) * -0.25)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(-x), (1.0d0 + ((x ** 2.0d0) * (-0.25d0))))
end function

Python

def code(x):
	return math.fmod(math.exp(-x), (1.0 + (math.pow(x, 2.0) * -0.25)))

Julia

function code(x)
	return rem(exp(Float64(-x)), Float64(1.0 + Float64((x ^ 2.0) * -0.25)))
end

Wolfram

code[x_] := N[With[{TMP1 = N[Exp[(-x)], $MachinePrecision], TMP2 = N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, 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. Add Preprocessing

3. Taylor expanded in x around 0 4.2%

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

   1. add-sqr-sqrt 2.3%

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

   2. sqrt-unprod 3.7%

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

   3. sqr-neg 3.7%

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

   4. sqrt-unprod 1.4%

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

   5. add-sqr-sqrt 23.5%

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

   6. neg-mul-1 23.5%

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

   7. exp-prod 23.5%

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

   8. add-log-exp 23.5%

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

   9. add-sqr-sqrt 23.5%

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

   10. log-prod 23.5%

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

   11. unpow-prod-up 23.5%

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

   12. pow1/2 23.5%

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

   13. log-pow 23.5%

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

   14. add-log-exp 23.5%

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

   15. pow1/2 23.5%

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

   16. log-pow 23.5%

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

   17. add-log-exp 23.5%

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

5. Applied egg-rr 23.5%

   \[\leadsto \left(\color{blue}{\left({\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)} \cdot {\left(e^{-1}\right)}^{\left(0.5 \cdot x\right)}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \]
6. Step-by-step derivation

   1. pow-sqr 23.5%

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

   2. associate-*r* 23.5%

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

   3. metadata-eval 23.5%

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

   4. *-lft-identity 23.5%

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

   5. exp-prod 23.5%

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

   6. neg-mul-1 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in x around 0 23.5%

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

9. Final simplification 23.5%

   \[\leadsto \left(\left(e^{-x}\right) \bmod \left(1 + {x}^{2} \cdot -0.25\right)\right) \]
10. Add Preprocessing

Alternative 6: 23.0% 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 4.8%

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

3. Taylor expanded in x around 0 4.2%

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

4. Taylor expanded in x around 0 4.2%

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

5. Taylor expanded in x around 0 4.8%

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

6. Taylor expanded in x around 0 23.3%

   \[\leadsto \left(1 \bmod \color{blue}{1}\right) \]
7. 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))))