expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 44.6%

Time: 29.4s

Alternatives: 4

Speedup: 505.0×

• could not determine a ground truth

  1. x = -7.913942824842468e+254

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: 44.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{{\left(\sqrt[3]{\log t_0 - x}\right)}^{2}}\right)\right)\right)}^{\left(\sqrt[3]{\log \left(\left(1 + t_0\right) + -1\right) - x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= x -5e-310)
     1.0
     (if (<= x 200.0)
       (pow
        (expm1 (log1p (exp (pow (cbrt (- (log t_0) x)) 2.0))))
        (cbrt (- (log (+ (+ 1.0 t_0) -1.0)) x)))
       1.0))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else if (x <= 200.0) {
		tmp = pow(expm1(log1p(exp(pow(cbrt((log(t_0) - x)), 2.0)))), cbrt((log(((1.0 + t_0) + -1.0)) - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = 1.0;
	elseif (x <= 200.0)
		tmp = expm1(log1p(exp((cbrt(Float64(log(t_0) - x)) ^ 2.0)))) ^ cbrt(Float64(log(Float64(Float64(1.0 + t_0) + -1.0)) - x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, If[LessEqual[x, -5e-310], 1.0, If[LessEqual[x, 200.0], N[Power[N[(Exp[N[Log[1 + N[Exp[N[Power[N[Power[N[(N[Log[t$95$0], $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[Power[N[(N[Log[N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 200 < x

   1. Initial program 8.7%

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

      1. exp-neg 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

   3. Simplified 8.7%

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

      1. add-exp-log 8.7%

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

      2. div-exp 8.8%

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

   5. Applied egg-rr 8.8%

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

      1. add-cube-cbrt 8.8%

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

      2. exp-prod 8.8%

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

      3. pow2 8.8%

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

   7. Applied egg-rr 8.8%

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

   8. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < x < 200

   1. Initial program 10.2%

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

      1. exp-neg 10.2%

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

      2. associate-*r/ 10.2%

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

      3. *-rgt-identity 10.2%

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

   3. Simplified 10.2%

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

      1. add-exp-log 10.2%

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

      2. div-exp 10.2%

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

   5. Applied egg-rr 10.2%

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

      1. add-cube-cbrt 10.1%

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

      2. exp-prod 10.1%

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

      3. pow2 10.1%

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

   7. Applied egg-rr 10.1%

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

      1. expm1-log1p-u 10.1%

         \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)} - x}\right)} \]

      2. expm1-udef 10.2%

         \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} - 1\right)} - x}\right)} \]

      3. log1p-udef 10.2%

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

      4. add-exp-log 10.2%

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

   9. Applied egg-rr 10.2%

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

       1. expm1-log1p-u 10.2%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)\right)\right)}}^{\left(\sqrt[3]{\log \left(\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - 1\right) - x}\right)} \]

   11. Applied egg-rr 10.2%

       \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)\right)\right)}}^{\left(\sqrt[3]{\log \left(\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - 1\right) - x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)\right)\right)}^{\left(\sqrt[3]{\log \left(\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + -1\right) - x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 44.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;{\left(e^{{\left(\sqrt[3]{\log t_0 - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\left(1 + t_0\right) + -1\right) - x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= x -5e-310)
     1.0
     (if (<= x 200.0)
       (pow
        (exp (pow (cbrt (- (log t_0) x)) 2.0))
        (cbrt (- (log (+ (+ 1.0 t_0) -1.0)) x)))
       1.0))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else if (x <= 200.0) {
		tmp = pow(exp(pow(cbrt((log(t_0) - x)), 2.0)), cbrt((log(((1.0 + t_0) + -1.0)) - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = 1.0;
	elseif (x <= 200.0)
		tmp = exp((cbrt(Float64(log(t_0) - x)) ^ 2.0)) ^ cbrt(Float64(log(Float64(Float64(1.0 + t_0) + -1.0)) - x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, If[LessEqual[x, -5e-310], 1.0, If[LessEqual[x, 200.0], N[Power[N[Exp[N[Power[N[Power[N[(N[Log[t$95$0], $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[Power[N[(N[Log[N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 200 < x

   1. Initial program 8.7%

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

      1. exp-neg 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

   3. Simplified 8.7%

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

      1. add-exp-log 8.7%

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

      2. div-exp 8.8%

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

   5. Applied egg-rr 8.8%

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

      1. add-cube-cbrt 8.8%

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

      2. exp-prod 8.8%

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

      3. pow2 8.8%

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

   7. Applied egg-rr 8.8%

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

   8. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < x < 200

   1. Initial program 10.2%

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

      1. exp-neg 10.2%

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

      2. associate-*r/ 10.2%

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

      3. *-rgt-identity 10.2%

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

   3. Simplified 10.2%

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

      1. add-exp-log 10.2%

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

      2. div-exp 10.2%

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

   5. Applied egg-rr 10.2%

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

      1. add-cube-cbrt 10.1%

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

      2. exp-prod 10.1%

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

      3. pow2 10.1%

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

   7. Applied egg-rr 10.1%

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

      1. expm1-log1p-u 10.1%

         \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)} - x}\right)} \]

      2. expm1-udef 10.2%

         \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} - 1\right)} - x}\right)} \]

      3. log1p-udef 10.2%

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

      4. add-exp-log 10.2%

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

   9. Applied egg-rr 10.2%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;{\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) + -1\right) - x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 44.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   1.0
   (if (<= x 200.0) (/ (fmod (exp x) (sqrt (cos x))) (exp x)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else if (x <= 200.0) {
		tmp = fmod(exp(x), sqrt(cos(x))) / exp(x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = 1.0d0
    else if (x <= 200.0d0) then
        tmp = mod(exp(x), sqrt(cos(x))) / exp(x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = 1.0
	elif x <= 200.0:
		tmp = math.fmod(math.exp(x), math.sqrt(math.cos(x))) / math.exp(x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = 1.0;
	elseif (x <= 200.0)
		tmp = Float64(rem(exp(x), sqrt(cos(x))) / exp(x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], 1.0, If[LessEqual[x, 200.0], 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], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310 or 200 < x

   1. Initial program 8.7%

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

      1. exp-neg 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

   3. Simplified 8.7%

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

      1. add-exp-log 8.7%

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

      2. div-exp 8.8%

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

   5. Applied egg-rr 8.8%

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

      1. add-cube-cbrt 8.8%

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

      2. exp-prod 8.8%

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

      3. pow2 8.8%

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

   7. Applied egg-rr 8.8%

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

   8. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < x < 200

   1. Initial program 10.2%

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

      1. exp-neg 10.2%

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

      2. associate-*r/ 10.2%

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

      3. *-rgt-identity 10.2%

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

   3. Simplified 10.2%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 43.4% 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 9.2%

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

   1. exp-neg 9.3%

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

   2. associate-*r/ 9.3%

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

   3. *-rgt-identity 9.3%

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

3. Simplified 9.3%

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

   1. add-exp-log 9.3%

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

   2. div-exp 9.3%

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

5. Applied egg-rr 9.3%

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

   1. add-cube-cbrt 9.3%

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

   2. exp-prod 9.3%

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

   3. pow2 9.3%

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

7. Applied egg-rr 9.3%

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

8. Taylor expanded in x around inf 46.7%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 46.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))