expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 62.2%

Time: 15.4s

Alternatives: 7

Speedup: 505.0×

• could not determine a ground truth

  1. x = -1.1340968397154998e+138

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 62.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\cos x}}\\ t_1 := {\left(\sqrt[3]{x}\right)}^{2}\\ \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{\left(\left({\left({\left(e^{t\_1}\right)}^{\left(\sqrt[3]{t\_1}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{x}}\right)}\right) \bmod \left(\sqrt{\log \left({t\_0}^{2}\right) + \log t\_0}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (cos x)))) (t_1 (pow (cbrt x) 2.0)))
   (if (<= x 500.0)
     (/
      (fmod
       (pow (pow (exp t_1) (cbrt t_1)) (cbrt (cbrt x)))
       (sqrt (+ (log (pow t_0 2.0)) (log t_0))))
      (exp x))
     (exp (- x)))))

C

double code(double x) {
	double t_0 = cbrt(exp(cos(x)));
	double t_1 = pow(cbrt(x), 2.0);
	double tmp;
	if (x <= 500.0) {
		tmp = fmod(pow(pow(exp(t_1), cbrt(t_1)), cbrt(cbrt(x))), sqrt((log(pow(t_0, 2.0)) + log(t_0)))) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(exp(cos(x)))
	t_1 = cbrt(x) ^ 2.0
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(rem(((exp(t_1) ^ cbrt(t_1)) ^ cbrt(cbrt(x))), sqrt(Float64(log((t_0 ^ 2.0)) + log(t_0)))) / exp(x));
	else
		tmp = 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]}, Block[{t$95$1 = N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, 500.0], N[(N[With[{TMP1 = N[Power[N[Power[N[Exp[t$95$1], $MachinePrecision], N[Power[t$95$1, 1/3], $MachinePrecision]], $MachinePrecision], N[Power[N[Power[x, 1/3], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], 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[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 9.8%

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

      1. /-rgt-identity 9.8%

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

      2. associate-/r/ 9.8%

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

      3. exp-neg 9.8%

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

      4. remove-double-neg 9.8%

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

   3. Simplified 9.8%

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

      1. add-log-exp 9.8%

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

      2. add-cube-cbrt 58.5%

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

      3. log-prod 58.5%

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

      4. pow2 58.5%

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

   6. Applied egg-rr 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. exp-prod 58.5%

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

      3. add-cube-cbrt 58.5%

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

      4. cbrt-prod 58.5%

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

      5. pow-unpow 58.5%

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

      6. pow2 58.5%

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

      7. pow2 58.5%

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

   8. Applied egg-rr 58.5%

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

   if 500 < x

   1. Initial program 0.0%

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

      1. exp-neg 0.0%

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

      2. div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. inv-pow 0.0%

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

      5. pow-to-exp 0.0%

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

      6. diff-log 0.0%

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

      7. add-log-exp 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   6. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\color{blue}{-x}} \]

   7. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\cos x}}\\ \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left({t\_0}^{2}\right) + \log t\_0}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (cos x)))))
   (if (<= x 350.0)
     (/ (fmod (exp x) (sqrt (+ (log (pow t_0 2.0)) (log t_0)))) (exp x))
     (exp (* (pow (cbrt x) 2.0) (- (cbrt x)))))))

C

double code(double x) {
	double t_0 = cbrt(exp(cos(x)));
	double tmp;
	if (x <= 350.0) {
		tmp = fmod(exp(x), sqrt((log(pow(t_0, 2.0)) + log(t_0)))) / exp(x);
	} else {
		tmp = exp((pow(cbrt(x), 2.0) * -cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(exp(cos(x)))
	tmp = 0.0
	if (x <= 350.0)
		tmp = Float64(rem(exp(x), sqrt(Float64(log((t_0 ^ 2.0)) + log(t_0)))) / exp(x));
	else
		tmp = exp(Float64((cbrt(x) ^ 2.0) * Float64(-cbrt(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, 350.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], 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[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 9.7%

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

      1. /-rgt-identity 9.7%

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

      2. associate-/r/ 9.7%

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

      3. exp-neg 9.7%

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

      4. remove-double-neg 9.7%

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

   3. Simplified 9.7%

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

      1. add-log-exp 9.7%

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

      2. add-cube-cbrt 58.7%

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

      3. log-prod 58.7%

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

      4. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

   if 350 < x

   1. Initial program 0.5%

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

      1. exp-neg 0.5%

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

      2. div-inv 0.5%

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

      3. clear-num 0.5%

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

      4. inv-pow 0.5%

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

      5. pow-to-exp 0.5%

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

      6. diff-log 0.5%

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

      7. add-log-exp 0.5%

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

   4. Applied egg-rr 0.5%

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

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   6. Step-by-step derivation

      1. neg-mul-1 98.2%

         \[\leadsto e^{\color{blue}{-x}} \]

   7. Simplified 98.2%

      \[\leadsto e^{\color{blue}{-x}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 98.2%

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

      2. distribute-rgt-neg-in 98.2%

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

      3. pow2 98.2%

         \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\sqrt[3]{x}\right)} \]

   9. Applied egg-rr 98.2%

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

Alternative 3: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod t\_0\right) \leq 10^{-156}:\\ \;\;\;\;e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(x + 1\right) \bmod t\_0\right) - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= (* (exp (- x)) (fmod (exp x) t_0)) 1e-156)
     (exp (* (pow (cbrt x) 2.0) (- (cbrt x))))
     (exp (- (log (fmod (+ x 1.0) t_0)) x)))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if ((exp(-x) * fmod(exp(x), t_0)) <= 1e-156) {
		tmp = exp((pow(cbrt(x), 2.0) * -cbrt(x)));
	} else {
		tmp = exp((log(fmod((x + 1.0), t_0)) - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), t_0)) <= 1e-156)
		tmp = exp(Float64((cbrt(x) ^ 2.0) * Float64(-cbrt(x))));
	else
		tmp = exp(Float64(log(rem(Float64(x + 1.0), t_0)) - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 1e-156], N[Exp[N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Log[N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1.00000000000000004e-156

   1. Initial program 4.2%

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

      1. exp-neg 4.2%

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

      2. div-inv 4.2%

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

      3. clear-num 4.2%

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

      4. inv-pow 4.2%

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

      5. pow-to-exp 4.2%

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

      6. diff-log 4.2%

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

      7. add-log-exp 4.2%

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

   4. Applied egg-rr 4.2%

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

   5. Taylor expanded in x around inf 57.3%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   6. Step-by-step derivation

      1. neg-mul-1 57.3%

         \[\leadsto e^{\color{blue}{-x}} \]

   7. Simplified 57.3%

      \[\leadsto e^{\color{blue}{-x}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 57.3%

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

      2. distribute-rgt-neg-in 57.3%

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

      3. pow2 57.3%

         \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\sqrt[3]{x}\right)} \]

   9. Applied egg-rr 57.3%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}} \]

   if 1.00000000000000004e-156 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 20.7%

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

      1. exp-neg 20.7%

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

      2. div-inv 20.7%

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

      3. clear-num 20.7%

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

      4. inv-pow 20.7%

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

      5. pow-to-exp 20.7%

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

      6. diff-log 20.7%

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

      7. add-log-exp 20.7%

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

   4. Applied egg-rr 20.7%

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

   5. Taylor expanded in x around 0 90.2%

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

      1. +-commutative 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-156}:\\ \;\;\;\;e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod t\_0\right) \leq 10^{-156}:\\ \;\;\;\;e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod t\_0\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (if (<= (* (exp (- x)) (fmod (exp x) t_0)) 1e-156)
     (exp (* (pow (cbrt x) 2.0) (- (cbrt x))))
     (/ (fmod (+ x 1.0) t_0) (exp x)))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double tmp;
	if ((exp(-x) * fmod(exp(x), t_0)) <= 1e-156) {
		tmp = exp((pow(cbrt(x), 2.0) * -cbrt(x)));
	} else {
		tmp = fmod((x + 1.0), t_0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	tmp = 0.0
	if (Float64(exp(Float64(-x)) * rem(exp(x), t_0)) <= 1e-156)
		tmp = exp(Float64((cbrt(x) ^ 2.0) * Float64(-cbrt(x))));
	else
		tmp = Float64(rem(Float64(x + 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[N[(N[Exp[(-x)], $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 1e-156], N[Exp[N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = t$95$0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1.00000000000000004e-156

   1. Initial program 4.2%

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

      1. exp-neg 4.2%

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

      2. div-inv 4.2%

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

      3. clear-num 4.2%

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

      4. inv-pow 4.2%

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

      5. pow-to-exp 4.2%

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

      6. diff-log 4.2%

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

      7. add-log-exp 4.2%

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

   4. Applied egg-rr 4.2%

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

   5. Taylor expanded in x around inf 57.3%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   6. Step-by-step derivation

      1. neg-mul-1 57.3%

         \[\leadsto e^{\color{blue}{-x}} \]

   7. Simplified 57.3%

      \[\leadsto e^{\color{blue}{-x}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 57.3%

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

      2. distribute-rgt-neg-in 57.3%

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

      3. pow2 57.3%

         \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\sqrt[3]{x}\right)} \]

   9. Applied egg-rr 57.3%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}} \]

   if 1.00000000000000004e-156 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 20.7%

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

      1. /-rgt-identity 20.7%

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

      2. associate-/r/ 20.6%

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

      3. exp-neg 20.7%

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

      4. remove-double-neg 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in x around 0 90.2%

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

      1. +-commutative 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \leq 10^{-156}:\\ \;\;\;\;e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ e^{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (* (pow (cbrt x) 2.0) (- (cbrt x)))))

C

double code(double x) {
	return exp((pow(cbrt(x), 2.0) * -cbrt(x)));
}

Java

public static double code(double x) {
	return Math.exp((Math.pow(Math.cbrt(x), 2.0) * -Math.cbrt(x)));
}

Julia

function code(x)
	return exp(Float64((cbrt(x) ^ 2.0) * Float64(-cbrt(x))))
end

Wolfram

code[x_] := N[Exp[N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 8.1%

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

   1. exp-neg 8.1%

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

   2. div-inv 8.1%

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

   3. clear-num 8.1%

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

   4. inv-pow 8.1%

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

   5. pow-to-exp 8.1%

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

   6. diff-log 8.1%

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

   7. add-log-exp 8.1%

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

4. Applied egg-rr 8.1%

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

5. Taylor expanded in x around inf 63.1%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
6. Step-by-step derivation

   1. neg-mul-1 63.1%

      \[\leadsto e^{\color{blue}{-x}} \]

7. Simplified 63.1%

   \[\leadsto e^{\color{blue}{-x}} \]
8. Step-by-step derivation

   1. add-cube-cbrt 63.1%

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

   2. distribute-rgt-neg-in 63.1%

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

   3. pow2 63.1%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\sqrt[3]{x}\right)} \]

9. Applied egg-rr 63.1%

   \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(-\sqrt[3]{x}\right)}} \]
10. Add Preprocessing

Alternative 6: 60.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

code[x_] := N[Exp[(-x)], $MachinePrecision]

Derivation

1. Initial program 8.1%

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

   1. exp-neg 8.1%

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

   2. div-inv 8.1%

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

   3. clear-num 8.1%

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

   4. inv-pow 8.1%

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

   5. pow-to-exp 8.1%

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

   6. diff-log 8.1%

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

   7. add-log-exp 8.1%

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

4. Applied egg-rr 8.1%

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

5. Taylor expanded in x around inf 63.1%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
6. Step-by-step derivation

   1. neg-mul-1 63.1%

      \[\leadsto e^{\color{blue}{-x}} \]

7. Simplified 63.1%

   \[\leadsto e^{\color{blue}{-x}} \]
8. Add Preprocessing

Alternative 7: 43.3% accurate, 505.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 8.1%

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

   1. exp-neg 8.1%

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

   2. div-inv 8.1%

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

   3. clear-num 8.1%

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

   4. inv-pow 8.1%

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

   5. pow-to-exp 8.1%

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

   6. diff-log 8.1%

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

   7. add-log-exp 8.1%

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

4. Applied egg-rr 8.1%

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

5. Taylor expanded in x around inf 63.1%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
6. Step-by-step derivation

   1. neg-mul-1 63.1%

      \[\leadsto e^{\color{blue}{-x}} \]

7. Simplified 63.1%

   \[\leadsto e^{\color{blue}{-x}} \]

8. Taylor expanded in x around 0 47.0%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))