expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 99.1%

Time: 19.7s

Alternatives: 9

Speedup: 5.0×

• could not determine a ground truth

  1. x = -1.812987445082115e+43

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.0% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\cos x}}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{x + 1}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{\log \left({t\_0}^{2}\right) + \log t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (cos x)))))
   (if (<= x -1e-8)
     (/ (fmod (+ x 1.0) (+ 1.0 (* -0.25 (pow x 2.0)))) (+ x 1.0))
     (if (<= x -5e-310)
       (fmod 1.0 (sqrt (+ (log (pow t_0 2.0)) (log t_0))))
       (* (fmod x (sqrt (cos x))) (exp (- x)))))))

C

double code(double x) {
	double t_0 = cbrt(exp(cos(x)));
	double tmp;
	if (x <= -1e-8) {
		tmp = fmod((x + 1.0), (1.0 + (-0.25 * pow(x, 2.0)))) / (x + 1.0);
	} else if (x <= -5e-310) {
		tmp = fmod(1.0, sqrt((log(pow(t_0, 2.0)) + log(t_0))));
	} else {
		tmp = fmod(x, sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(exp(cos(x)))
	tmp = 0.0
	if (x <= -1e-8)
		tmp = Float64(rem(Float64(x + 1.0), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / Float64(x + 1.0));
	elseif (x <= -5e-310)
		tmp = rem(1.0, sqrt(Float64(log((t_0 ^ 2.0)) + log(t_0))));
	else
		tmp = Float64(rem(x, sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, -1e-8], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-310], N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = x, 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 3 regimes

2. if x < -1e-8

   1. Initial program 59.7%

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

      1. /-rgt-identity 59.7%

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

      2. associate-/r/ 59.7%

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

      3. exp-neg 60.0%

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

      4. remove-double-neg 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in x around 0 30.6%

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

      1. +-commutative 30.6%

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

   7. Simplified 30.6%

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

   8. Taylor expanded in x around 0 30.6%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. +-commutative 30.6%

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

   11. Simplified 100.0%

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

   if -1e-8 < x < -4.999999999999985e-310

   1. Initial program 4.9%

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

      1. /-rgt-identity 4.9%

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

      2. associate-/r/ 4.9%

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

      3. exp-neg 4.9%

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

      4. remove-double-neg 4.9%

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

   3. Simplified 4.9%

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

   5. Taylor expanded in x around 0 4.4%

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

   6. Taylor expanded in x around 0 3.1%

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

      1. add-log-exp 4.9%

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

      2. add-cube-cbrt 100.0%

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

      3. log-prod 100.0%

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

      4. pow2 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.4%

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

      1. /-rgt-identity 5.4%

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

      2. associate-/r/ 5.4%

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

      3. exp-neg 5.4%

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

      4. remove-double-neg 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around 0 43.3%

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

      1. +-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in x around inf 98.1%

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

      1. div-inv 98.1%

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

      2. rec-exp 98.1%

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

   10. Applied egg-rr 98.1%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({\left(\sqrt[3]{e^{\cos x}}\right)}^{2}\right) + \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/
    (fmod
     (exp x)
     (sqrt (+ (log (pow (cbrt (exp (cos x))) 2.0)) (log (cbrt E)))))
    (exp x))
   (* (fmod x (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), sqrt((log(pow(cbrt(exp(cos(x))), 2.0)) + log(cbrt(((double) M_E)))))) / exp(x);
	} else {
		tmp = fmod(x, sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), sqrt(Float64(log((cbrt(exp(cos(x))) ^ 2.0)) + log(cbrt(exp(1)))))) / exp(x));
	else
		tmp = Float64(rem(x, sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 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 = x, 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 < -4.999999999999985e-310

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

      1. add-log-exp 7.4%

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

      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%

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

   if -4.999999999999985e-310 < x

   1. Initial program 5.4%

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

      1. /-rgt-identity 5.4%

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

      2. associate-/r/ 5.4%

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

      3. exp-neg 5.4%

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

      4. remove-double-neg 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around 0 43.3%

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

      1. +-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in x around inf 98.1%

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

      1. div-inv 98.1%

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

      2. rec-exp 98.1%

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

   10. Applied egg-rr 98.1%

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

Alternative 3: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(x \bmod t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod t\_0\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= x -6e-309)
     (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x))
     (if (<= x 1.0) (fmod x t_0) (/ (fmod 1.0 t_0) (exp x))))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
	} else if (x <= 1.0) {
		tmp = fmod(x, t_0);
	} else {
		tmp = fmod(1.0, t_0) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(cos(x))
    if (x <= (-6d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
    else if (x <= 1.0d0) then
        tmp = mod(x, t_0)
    else
        tmp = mod(1.0d0, t_0) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.sqrt(math.cos(x))
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)
	elif x <= 1.0:
		tmp = math.fmod(x, t_0)
	else:
		tmp = math.fmod(1.0, t_0) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x));
	elseif (x <= 1.0)
		tmp = rem(x, t_0);
	else
		tmp = Float64(rem(1.0, t_0) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[With[{TMP1 = x, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.000000000000001e-309

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 6.1%

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

   9. Taylor expanded in x around inf 12.9%

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

   if -6.000000000000001e-309 < x < 1

   1. Initial program 7.7%

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

      1. /-rgt-identity 7.7%

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

      2. associate-/r/ 7.7%

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

      3. exp-neg 7.7%

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

      4. remove-double-neg 7.7%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in x around inf 98.6%

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

   9. Taylor expanded in x around 0 98.7%

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

   if 1 < x

   1. Initial program 2.1%

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

      1. /-rgt-identity 2.1%

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

      2. associate-/r/ 2.1%

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

      3. exp-neg 2.1%

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

      4. remove-double-neg 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around 0 97.2%

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

Alternative 4: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x))
   (* (fmod x (sqrt (cos x))) (exp (- x)))))

C

double code(double x) {
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
	} else {
		tmp = fmod(x, sqrt(cos(x))) * exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
    else
        tmp = mod(x, sqrt(cos(x))) * exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)
	else:
		tmp = math.fmod(x, math.sqrt(math.cos(x))) * math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x));
	else
		tmp = Float64(rem(x, sqrt(cos(x))) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, 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 < -6.000000000000001e-309

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 6.1%

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

   9. Taylor expanded in x around inf 12.9%

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

   if -6.000000000000001e-309 < x

   1. Initial program 5.4%

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

      1. /-rgt-identity 5.4%

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

      2. associate-/r/ 5.4%

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

      3. exp-neg 5.4%

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

      4. remove-double-neg 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around 0 43.3%

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

      1. +-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in x around inf 98.1%

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

      1. div-inv 98.1%

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

      2. rec-exp 98.1%

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

   10. Applied egg-rr 98.1%

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

Alternative 5: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x))
   (/ (fmod x (sqrt (cos x))) (exp x))))

C

double code(double x) {
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
	} else {
		tmp = fmod(x, sqrt(cos(x))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
    else
        tmp = mod(x, sqrt(cos(x))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)
	else:
		tmp = math.fmod(x, math.sqrt(math.cos(x))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x));
	else
		tmp = Float64(rem(x, sqrt(cos(x))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = x, 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 < -6.000000000000001e-309

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 6.1%

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

   9. Taylor expanded in x around inf 12.9%

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

   if -6.000000000000001e-309 < x

   1. Initial program 5.4%

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

      1. /-rgt-identity 5.4%

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

      2. associate-/r/ 5.4%

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

      3. exp-neg 5.4%

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

      4. remove-double-neg 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around 0 43.3%

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

      1. +-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in x around inf 98.1%

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

Alternative 6: 66.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}}\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-309)
   (/ (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x))
   (if (<= x 0.92)
     (fmod x (sqrt (cos x)))
     (/ (fmod (+ x 1.0) (+ 1.0 (* -0.25 (* x x)))) (exp x)))))

C

double code(double x) {
	double tmp;
	if (x <= -6e-309) {
		tmp = fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
	} else if (x <= 0.92) {
		tmp = fmod(x, sqrt(cos(x)));
	} else {
		tmp = fmod((x + 1.0), (1.0 + (-0.25 * (x * x)))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6d-309)) then
        tmp = mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
    else if (x <= 0.92d0) then
        tmp = mod(x, sqrt(cos(x)))
    else
        tmp = mod((x + 1.0d0), (1.0d0 + ((-0.25d0) * (x * x)))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -6e-309:
		tmp = math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)
	elif x <= 0.92:
		tmp = math.fmod(x, math.sqrt(math.cos(x)))
	else:
		tmp = math.fmod((x + 1.0), (1.0 + (-0.25 * (x * x)))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-309)
		tmp = Float64(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x));
	elseif (x <= 0.92)
		tmp = rem(x, sqrt(cos(x)));
	else
		tmp = Float64(rem(Float64(x + 1.0), Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6e-309], N[(N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.92], N[With[{TMP1 = x, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.000000000000001e-309

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 6.1%

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

   9. Taylor expanded in x around inf 12.9%

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

   if -6.000000000000001e-309 < x < 0.92000000000000004

   1. Initial program 7.7%

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

      1. /-rgt-identity 7.7%

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

      2. associate-/r/ 7.7%

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

      3. exp-neg 7.7%

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

      4. remove-double-neg 7.7%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in x around inf 98.6%

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

   9. Taylor expanded in x around 0 98.7%

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

   if 0.92000000000000004 < x

   1. Initial program 2.1%

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

      1. /-rgt-identity 2.1%

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

      2. associate-/r/ 2.1%

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

      3. exp-neg 2.1%

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

      4. remove-double-neg 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around 0 97.2%

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

      1. +-commutative 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0 97.1%

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

      1. unpow2 97.1%

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

   10. Applied egg-rr 97.1%

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

Alternative 7: 62.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{x + 1}\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;\left(x \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/ (fmod (+ x 1.0) (+ 1.0 (* -0.25 (pow x 2.0)))) (+ x 1.0))
   (if (<= x 0.92)
     (fmod x (sqrt (cos x)))
     (/ (fmod (+ x 1.0) (+ 1.0 (* -0.25 (* x x)))) (exp x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod((x + 1.0), (1.0 + (-0.25 * pow(x, 2.0)))) / (x + 1.0);
	} else if (x <= 0.92) {
		tmp = fmod(x, sqrt(cos(x)));
	} else {
		tmp = fmod((x + 1.0), (1.0 + (-0.25 * (x * x)))) / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = mod((x + 1.0d0), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / (x + 1.0d0)
    else if (x <= 0.92d0) then
        tmp = mod(x, sqrt(cos(x)))
    else
        tmp = mod((x + 1.0d0), (1.0d0 + ((-0.25d0) * (x * x)))) / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.fmod((x + 1.0), (1.0 + (-0.25 * math.pow(x, 2.0)))) / (x + 1.0)
	elif x <= 0.92:
		tmp = math.fmod(x, math.sqrt(math.cos(x)))
	else:
		tmp = math.fmod((x + 1.0), (1.0 + (-0.25 * (x * x)))) / math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(Float64(x + 1.0), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / Float64(x + 1.0));
	elseif (x <= 0.92)
		tmp = rem(x, sqrt(cos(x)));
	else
		tmp = Float64(rem(Float64(x + 1.0), Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.92], N[With[{TMP1 = x, TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 7.4%

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

      1. /-rgt-identity 7.4%

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

      2. associate-/r/ 7.4%

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

      3. exp-neg 7.4%

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

      4. remove-double-neg 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 6.1%

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

   9. Taylor expanded in x around 0 9.2%

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

       1. +-commutative 6.1%

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

   11. Simplified 9.2%

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

   if -4.999999999999985e-310 < x < 0.92000000000000004

   1. Initial program 7.7%

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

      1. /-rgt-identity 7.7%

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

      2. associate-/r/ 7.7%

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

      3. exp-neg 7.7%

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

      4. remove-double-neg 7.7%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in x around inf 98.6%

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

   9. Taylor expanded in x around 0 98.7%

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

   if 0.92000000000000004 < x

   1. Initial program 2.1%

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

      1. /-rgt-identity 2.1%

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

      2. associate-/r/ 2.1%

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

      3. exp-neg 2.1%

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

      4. remove-double-neg 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around 0 97.2%

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

      1. +-commutative 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0 97.1%

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

      1. unpow2 97.1%

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

   10. Applied egg-rr 97.1%

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

Alternative 8: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. /-rgt-identity 6.3%

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

   2. associate-/r/ 6.3%

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

   3. exp-neg 6.3%

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

   4. remove-double-neg 6.3%

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

3. Simplified 6.3%

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

5. Taylor expanded in x around 0 27.0%

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

   1. +-commutative 27.0%

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

7. Simplified 27.0%

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

8. Taylor expanded in x around 0 27.0%

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

   1. unpow2 27.0%

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

10. Applied egg-rr 27.0%

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

Alternative 9: 22.8% 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 6.3%

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

   1. /-rgt-identity 6.3%

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

   2. associate-/r/ 6.3%

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

   3. exp-neg 6.3%

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

   4. remove-double-neg 6.3%

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

3. Simplified 6.3%

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

5. Taylor expanded in x around 0 4.7%

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

6. Taylor expanded in x around 0 4.3%

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

7. Taylor expanded in x around 0 5.0%

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

8. Taylor expanded in x around 0 25.2%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))