expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 12.7%

Time: 18.6s

Alternatives: 16

Speedup: 1.0×

• could not determine a ground truth

  1. x = -1.2806673614436495e+271

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 12.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \mathsf{expm1}\left(t_0 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := 1 - t_1\\ t_4 := {t_3}^{-0.5}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{t_3} - \frac{t_2}{1 - \mathsf{expm1}\left(\left|t_0\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_4, t_4, \frac{-t_2}{t_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (sqrt (cos x)))))
        (t_1 (expm1 (- t_0 x)))
        (t_2 (pow t_1 2.0))
        (t_3 (- 1.0 t_1))
        (t_4 (pow t_3 -0.5)))
   (if (<= x -2e-311)
     (- (/ 1.0 t_3) (/ t_2 (- 1.0 (expm1 (- (fabs t_0) x)))))
     (fma t_4 t_4 (/ (- t_2) t_3)))))

C

double code(double x) {
	double t_0 = log(fmod(exp(x), sqrt(cos(x))));
	double t_1 = expm1((t_0 - x));
	double t_2 = pow(t_1, 2.0);
	double t_3 = 1.0 - t_1;
	double t_4 = pow(t_3, -0.5);
	double tmp;
	if (x <= -2e-311) {
		tmp = (1.0 / t_3) - (t_2 / (1.0 - expm1((fabs(t_0) - x))));
	} else {
		tmp = fma(t_4, t_4, (-t_2 / t_3));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = log(rem(exp(x), sqrt(cos(x))))
	t_1 = expm1(Float64(t_0 - x))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(1.0 - t_1)
	t_4 = t_3 ^ -0.5
	tmp = 0.0
	if (x <= -2e-311)
		tmp = Float64(Float64(1.0 / t_3) - Float64(t_2 / Float64(1.0 - expm1(Float64(abs(t_0) - x)))));
	else
		tmp = fma(t_4, t_4, Float64(Float64(-t_2) / t_3));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Exp[N[(t$95$0 - x), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, -0.5], $MachinePrecision]}, If[LessEqual[x, -2e-311], N[(N[(1.0 / t$95$3), $MachinePrecision] - N[(t$95$2 / N[(1.0 - N[(Exp[N[(N[Abs[t$95$0], $MachinePrecision] - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * t$95$4 + N[((-t$95$2) / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 12.3%

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

      1. exp-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. *-rgt-identity 12.4%

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

   3. Simplified 12.4%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 12.4%

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

      3. log1p-udef 12.4%

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

      4. add-exp-log 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate--l+ 12.4%

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

   7. Simplified 12.4%

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

      1. flip-+ 12.4%

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

      2. metadata-eval 12.4%

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

      3. div-sub 12.4%

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

   9. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 26.6%

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

       3. pow2 26.6%

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

   11. Applied egg-rr 26.6%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 26.6%

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

       2. rem-sqrt-square 26.6%

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

   13. Simplified 26.6%

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

   if -1.9999999999999e-311 < x

   1. Initial program 6.8%

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

      1. exp-neg 6.8%

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

      2. associate-*r/ 6.8%

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

      3. *-rgt-identity 6.8%

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

   3. Simplified 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-udef 6.7%

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

      3. log1p-udef 6.7%

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

      4. add-exp-log 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. associate--l+ 6.8%

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

   7. Simplified 6.8%

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

      1. flip-+ 6.7%

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

      2. metadata-eval 6.7%

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

      3. div-sub 6.8%

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

   9. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 6.8%

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

       2. fma-neg 6.9%

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

   11. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{-0.5}, {\left(1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{-0.5}, -\frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{-0.5}, {\left(1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{-0.5}, \frac{-{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\right)\\ \end{array} \]

Alternative 2: 12.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \mathsf{expm1}\left(t_0 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := 1 - t_1\\ t_4 := \frac{1}{t_3}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t_4 - \frac{t_2}{1 - \mathsf{expm1}\left(\left|t_0\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4 - \sqrt[3]{{\left(\frac{t_2}{t_3}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (sqrt (cos x)))))
        (t_1 (expm1 (- t_0 x)))
        (t_2 (pow t_1 2.0))
        (t_3 (- 1.0 t_1))
        (t_4 (/ 1.0 t_3)))
   (if (<= x -2e-311)
     (- t_4 (/ t_2 (- 1.0 (expm1 (- (fabs t_0) x)))))
     (- t_4 (cbrt (pow (/ t_2 t_3) 3.0))))))

C

double code(double x) {
	double t_0 = log(fmod(exp(x), sqrt(cos(x))));
	double t_1 = expm1((t_0 - x));
	double t_2 = pow(t_1, 2.0);
	double t_3 = 1.0 - t_1;
	double t_4 = 1.0 / t_3;
	double tmp;
	if (x <= -2e-311) {
		tmp = t_4 - (t_2 / (1.0 - expm1((fabs(t_0) - x))));
	} else {
		tmp = t_4 - cbrt(pow((t_2 / t_3), 3.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = log(rem(exp(x), sqrt(cos(x))))
	t_1 = expm1(Float64(t_0 - x))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(1.0 - t_1)
	t_4 = Float64(1.0 / t_3)
	tmp = 0.0
	if (x <= -2e-311)
		tmp = Float64(t_4 - Float64(t_2 / Float64(1.0 - expm1(Float64(abs(t_0) - x)))));
	else
		tmp = Float64(t_4 - cbrt((Float64(t_2 / t_3) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Exp[N[(t$95$0 - x), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / t$95$3), $MachinePrecision]}, If[LessEqual[x, -2e-311], N[(t$95$4 - N[(t$95$2 / N[(1.0 - N[(Exp[N[(N[Abs[t$95$0], $MachinePrecision] - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 - N[Power[N[Power[N[(t$95$2 / t$95$3), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 12.3%

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

      1. exp-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. *-rgt-identity 12.4%

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

   3. Simplified 12.4%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 12.4%

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

      3. log1p-udef 12.4%

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

      4. add-exp-log 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate--l+ 12.4%

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

   7. Simplified 12.4%

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

      1. flip-+ 12.4%

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

      2. metadata-eval 12.4%

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

      3. div-sub 12.4%

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

   9. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 26.6%

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

       3. pow2 26.6%

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

   11. Applied egg-rr 26.6%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 26.6%

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

       2. rem-sqrt-square 26.6%

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

   13. Simplified 26.6%

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

   if -1.9999999999999e-311 < x

   1. Initial program 6.8%

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

      1. exp-neg 6.8%

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

      2. associate-*r/ 6.8%

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

      3. *-rgt-identity 6.8%

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

   3. Simplified 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-udef 6.7%

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

      3. log1p-udef 6.7%

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

      4. add-exp-log 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. associate--l+ 6.8%

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

   7. Simplified 6.8%

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

      1. flip-+ 6.7%

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

      2. metadata-eval 6.7%

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

      3. div-sub 6.8%

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

   9. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.9%

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

       3. pow2 3.9%

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

   11. Applied egg-rr 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 3.9%

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

       2. rem-sqrt-square 3.9%

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

   13. Simplified 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right|} - x\right)} \]
   14. Step-by-step derivation

       1. add-cbrt-cube 3.9%

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

       2. pow3 3.9%

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

   15. Applied egg-rr 6.9%

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

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \sqrt[3]{{\left(\frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\right)}^{3}}\\ \end{array} \]

Alternative 3: 12.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \log t_0\\ t_2 := \mathsf{expm1}\left(t_1 - x\right)\\ t_3 := \frac{1}{1 - t_2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t_3 - \frac{{t_2}^{2}}{1 - \mathsf{expm1}\left(\left|t_1\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3 - {\left(\frac{t_2}{\sqrt{2 - \frac{t_0}{e^{x}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (log t_0))
        (t_2 (expm1 (- t_1 x)))
        (t_3 (/ 1.0 (- 1.0 t_2))))
   (if (<= x -2e-311)
     (- t_3 (/ (pow t_2 2.0) (- 1.0 (expm1 (- (fabs t_1) x)))))
     (- t_3 (pow (/ t_2 (sqrt (- 2.0 (/ t_0 (exp x))))) 2.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = log(t_0);
	double t_2 = expm1((t_1 - x));
	double t_3 = 1.0 / (1.0 - t_2);
	double tmp;
	if (x <= -2e-311) {
		tmp = t_3 - (pow(t_2, 2.0) / (1.0 - expm1((fabs(t_1) - x))));
	} else {
		tmp = t_3 - pow((t_2 / sqrt((2.0 - (t_0 / exp(x))))), 2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.log(t_0)
	t_2 = math.expm1((t_1 - x))
	t_3 = 1.0 / (1.0 - t_2)
	tmp = 0
	if x <= -2e-311:
		tmp = t_3 - (math.pow(t_2, 2.0) / (1.0 - math.expm1((math.fabs(t_1) - x))))
	else:
		tmp = t_3 - math.pow((t_2 / math.sqrt((2.0 - (t_0 / math.exp(x))))), 2.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = log(t_0)
	t_2 = expm1(Float64(t_1 - x))
	t_3 = Float64(1.0 / Float64(1.0 - t_2))
	tmp = 0.0
	if (x <= -2e-311)
		tmp = Float64(t_3 - Float64((t_2 ^ 2.0) / Float64(1.0 - expm1(Float64(abs(t_1) - x)))));
	else
		tmp = Float64(t_3 - (Float64(t_2 / sqrt(Float64(2.0 - Float64(t_0 / exp(x))))) ^ 2.0));
	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]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Exp[N[(t$95$1 - x), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-311], N[(t$95$3 - N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(1.0 - N[(Exp[N[(N[Abs[t$95$1], $MachinePrecision] - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 - N[Power[N[(t$95$2 / N[Sqrt[N[(2.0 - N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 12.3%

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

      1. exp-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. *-rgt-identity 12.4%

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

   3. Simplified 12.4%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 12.4%

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

      3. log1p-udef 12.4%

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

      4. add-exp-log 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate--l+ 12.4%

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

   7. Simplified 12.4%

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

      1. flip-+ 12.4%

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

      2. metadata-eval 12.4%

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

      3. div-sub 12.4%

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

   9. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 26.6%

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

       3. pow2 26.6%

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

   11. Applied egg-rr 26.6%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 26.6%

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

       2. rem-sqrt-square 26.6%

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

   13. Simplified 26.6%

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

   if -1.9999999999999e-311 < x

   1. Initial program 6.8%

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

      1. exp-neg 6.8%

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

      2. associate-*r/ 6.8%

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

      3. *-rgt-identity 6.8%

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

   3. Simplified 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-udef 6.7%

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

      3. log1p-udef 6.7%

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

      4. add-exp-log 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. associate--l+ 6.8%

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

   7. Simplified 6.8%

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

      1. flip-+ 6.7%

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

      2. metadata-eval 6.7%

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

      3. div-sub 6.8%

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

   9. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.9%

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

       3. pow2 3.9%

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

   11. Applied egg-rr 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 3.9%

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

       2. rem-sqrt-square 3.9%

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

   13. Simplified 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right|} - x\right)} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 3.4%

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

   15. Applied egg-rr 6.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \color{blue}{\frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \cdot \frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}}} \]
   16. Step-by-step derivation

       1. unpow2 6.9%

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

   17. Simplified 6.9%

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

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right| - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - {\left(\frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{2 - \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)}^{2}\\ \end{array} \]

Alternative 4: 12.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \log t_0\\ t_2 := \mathsf{expm1}\left(t_1 - x\right)\\ t_3 := \frac{1}{1 - t_2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t_3 - \frac{{\left(t_0 + -1\right)}^{2}}{2 - e^{\left|t_1\right|}}\\ \mathbf{else}:\\ \;\;\;\;t_3 - {\left(\frac{t_2}{\sqrt{2 - \frac{t_0}{e^{x}}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (log t_0))
        (t_2 (expm1 (- t_1 x)))
        (t_3 (/ 1.0 (- 1.0 t_2))))
   (if (<= x -2e-311)
     (- t_3 (/ (pow (+ t_0 -1.0) 2.0) (- 2.0 (exp (fabs t_1)))))
     (- t_3 (pow (/ t_2 (sqrt (- 2.0 (/ t_0 (exp x))))) 2.0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = log(t_0);
	double t_2 = expm1((t_1 - x));
	double t_3 = 1.0 / (1.0 - t_2);
	double tmp;
	if (x <= -2e-311) {
		tmp = t_3 - (pow((t_0 + -1.0), 2.0) / (2.0 - exp(fabs(t_1))));
	} else {
		tmp = t_3 - pow((t_2 / sqrt((2.0 - (t_0 / exp(x))))), 2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.log(t_0)
	t_2 = math.expm1((t_1 - x))
	t_3 = 1.0 / (1.0 - t_2)
	tmp = 0
	if x <= -2e-311:
		tmp = t_3 - (math.pow((t_0 + -1.0), 2.0) / (2.0 - math.exp(math.fabs(t_1))))
	else:
		tmp = t_3 - math.pow((t_2 / math.sqrt((2.0 - (t_0 / math.exp(x))))), 2.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = log(t_0)
	t_2 = expm1(Float64(t_1 - x))
	t_3 = Float64(1.0 / Float64(1.0 - t_2))
	tmp = 0.0
	if (x <= -2e-311)
		tmp = Float64(t_3 - Float64((Float64(t_0 + -1.0) ^ 2.0) / Float64(2.0 - exp(abs(t_1)))));
	else
		tmp = Float64(t_3 - (Float64(t_2 / sqrt(Float64(2.0 - Float64(t_0 / exp(x))))) ^ 2.0));
	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]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Exp[N[(t$95$1 - x), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-311], N[(t$95$3 - N[(N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 - N[Exp[N[Abs[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 - N[Power[N[(t$95$2 / N[Sqrt[N[(2.0 - N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 12.3%

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

      1. exp-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. *-rgt-identity 12.4%

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

   3. Simplified 12.4%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 12.4%

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

      3. log1p-udef 12.4%

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

      4. add-exp-log 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate--l+ 12.4%

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

   7. Simplified 12.4%

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

      1. flip-+ 12.4%

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

      2. metadata-eval 12.4%

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

      3. div-sub 12.4%

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

   9. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 26.6%

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

       3. pow2 26.6%

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

   11. Applied egg-rr 26.6%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 26.6%

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

       2. rem-sqrt-square 26.6%

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

   13. Simplified 26.6%

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

   14. Taylor expanded in x around 0 24.8%

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

   if -1.9999999999999e-311 < x

   1. Initial program 6.8%

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

      1. exp-neg 6.8%

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

      2. associate-*r/ 6.8%

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

      3. *-rgt-identity 6.8%

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

   3. Simplified 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-udef 6.7%

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

      3. log1p-udef 6.7%

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

      4. add-exp-log 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. associate--l+ 6.8%

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

   7. Simplified 6.8%

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

      1. flip-+ 6.7%

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

      2. metadata-eval 6.7%

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

      3. div-sub 6.8%

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

   9. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.9%

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

       3. pow2 3.9%

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

   11. Applied egg-rr 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 3.9%

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

       2. rem-sqrt-square 3.9%

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

   13. Simplified 3.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right|} - x\right)} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 3.4%

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

   15. Applied egg-rr 6.9%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \color{blue}{\frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \cdot \frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}}} \]
   16. Step-by-step derivation

       1. unpow2 6.9%

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

   17. Simplified 6.9%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + -1\right)}^{2}}{2 - e^{\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - {\left(\frac{\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}{\sqrt{2 - \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}}\right)}^{2}\\ \end{array} \]

Alternative 5: 12.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \log t_0\\ \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(t_1 - x\right)} - \frac{{\left(t_0 + -1\right)}^{2}}{2 - e^{\left|t_1\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{t_0}\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (log t_0)))
   (if (<= x -2e-311)
     (-
      (/ 1.0 (- 1.0 (expm1 (- t_1 x))))
      (/ (pow (+ t_0 -1.0) 2.0) (- 2.0 (exp (fabs t_1)))))
     (/ (log (exp t_0)) (exp x)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = log(t_0);
	double tmp;
	if (x <= -2e-311) {
		tmp = (1.0 / (1.0 - expm1((t_1 - x)))) - (pow((t_0 + -1.0), 2.0) / (2.0 - exp(fabs(t_1))));
	} else {
		tmp = log(exp(t_0)) / exp(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.log(t_0)
	tmp = 0
	if x <= -2e-311:
		tmp = (1.0 / (1.0 - math.expm1((t_1 - x)))) - (math.pow((t_0 + -1.0), 2.0) / (2.0 - math.exp(math.fabs(t_1))))
	else:
		tmp = math.log(math.exp(t_0)) / math.exp(x)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = log(t_0)
	tmp = 0.0
	if (x <= -2e-311)
		tmp = Float64(Float64(1.0 / Float64(1.0 - expm1(Float64(t_1 - x)))) - Float64((Float64(t_0 + -1.0) ^ 2.0) / Float64(2.0 - exp(abs(t_1)))));
	else
		tmp = Float64(log(exp(t_0)) / exp(x));
	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]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, If[LessEqual[x, -2e-311], N[(N[(1.0 / N[(1.0 - N[(Exp[N[(t$95$1 - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 - N[Exp[N[Abs[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 12.3%

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

      1. exp-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. *-rgt-identity 12.4%

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

   3. Simplified 12.4%

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

      1. expm1-log1p-u 12.4%

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

      2. expm1-udef 12.4%

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

      3. log1p-udef 12.4%

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

      4. add-exp-log 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate--l+ 12.4%

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

   7. Simplified 12.4%

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

      1. flip-+ 12.4%

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

      2. metadata-eval 12.4%

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

      3. div-sub 12.4%

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

   9. Applied egg-rr 12.4%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 26.6%

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

       3. pow2 26.6%

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

   11. Applied egg-rr 26.6%

       \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\color{blue}{\sqrt{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}}} - x\right)} \]
   12. Step-by-step derivation

       1. unpow2 26.6%

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

       2. rem-sqrt-square 26.6%

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

   13. Simplified 26.6%

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

   14. Taylor expanded in x around 0 24.8%

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

   if -1.9999999999999e-311 < x

   1. Initial program 6.8%

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

      1. exp-neg 6.8%

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

      2. associate-*r/ 6.8%

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

      3. *-rgt-identity 6.8%

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

   3. Simplified 6.8%

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

      1. add-log-exp 6.8%

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

   5. Applied egg-rr 6.8%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)} - \frac{{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + -1\right)}^{2}}{2 - e^{\left|\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}}\\ \end{array} \]

Alternative 6: 6.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)\\ \frac{{t_0}^{2} + -1}{t_0 + -1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (expm1 (- (log (fmod (exp x) (sqrt (cos x)))) x))))
   (/ (+ (pow t_0 2.0) -1.0) (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = expm1((log(fmod(exp(x), sqrt(cos(x)))) - x));
	return (pow(t_0, 2.0) + -1.0) / (t_0 + -1.0);
}

Python

def code(x):
	t_0 = math.expm1((math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x)))) - x))
	return (math.pow(t_0, 2.0) + -1.0) / (t_0 + -1.0)

Julia

function code(x)
	t_0 = expm1(Float64(log(rem(exp(x), sqrt(cos(x)))) - x))
	return Float64(Float64((t_0 ^ 2.0) + -1.0) / Float64(t_0 + -1.0))
end

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

5. Applied egg-rr 9.0%

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

   1. associate--l+ 9.0%

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

7. Simplified 9.0%

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

   1. +-commutative 9.0%

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

   2. flip-+ 9.0%

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

   3. pow2 9.0%

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

   4. add-exp-log 9.0%

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

   5. expm1-def 9.0%

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

   6. log-div 9.0%

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

   7. add-log-exp 9.0%

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

   8. metadata-eval 9.0%

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

9. Applied egg-rr 9.1%

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

10. Final simplification 9.1%

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

Alternative 7: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 + N[(1.0 + N[(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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

5. Applied egg-rr 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

7. Applied egg-rr 9.0%

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

   1. associate--l+ 9.0%

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

9. Simplified 9.0%

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

10. Final simplification 9.0%

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

Alternative 8: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(1.0 + N[(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), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

5. Applied egg-rr 9.0%

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

   1. associate--l+ 9.0%

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

7. Simplified 9.0%

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

8. Final simplification 9.0%

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

Alternative 9: 6.9% accurate, 1.0× speedup

Math

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

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(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]

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Final simplification 9.0%

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

Alternative 10: 6.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

5. Applied egg-rr 9.0%

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

   1. associate--l+ 9.0%

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

7. Simplified 9.0%

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

8. Taylor expanded in x around 0 8.7%

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

   1. *-commutative 8.7%

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

   2. unpow2 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Alternative 11: 6.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Taylor expanded in x around 0 8.7%

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

   1. *-commutative 8.7%

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

   2. unpow2 8.7%

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

6. Simplified 8.7%

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

7. Final simplification 8.7%

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

Alternative 12: 6.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

   1. expm1-log1p-u 9.0%

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

   2. expm1-udef 9.0%

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

   3. log1p-udef 9.0%

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

   4. add-exp-log 9.0%

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

5. Applied egg-rr 9.0%

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

   1. associate--l+ 9.0%

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

7. Simplified 9.0%

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

8. Taylor expanded in x around 0 8.2%

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

9. Final simplification 8.2%

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

Alternative 13: 6.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Taylor expanded in x around 0 8.2%

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

5. Final simplification 8.2%

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

Alternative 14: 5.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Taylor expanded in x around 0 8.2%

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

5. Taylor expanded in x around 0 7.4%

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

   1. +-commutative 7.4%

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

   2. *-lft-identity 7.4%

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

   3. associate-*r* 7.4%

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

   4. neg-mul-1 7.4%

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

   5. distribute-rgt-out 7.4%

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

   6. unsub-neg 7.4%

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

7. Simplified 7.4%

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

   1. flip-- 7.4%

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

   2. associate-*r/ 7.4%

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

   3. metadata-eval 7.4%

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

   4. +-commutative 7.4%

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

9. Applied egg-rr 7.4%

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

    1. associate-/l* 7.4%

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

11. Simplified 7.4%

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

12. Final simplification 7.4%

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

Alternative 15: 5.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Taylor expanded in x around 0 8.2%

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

5. Taylor expanded in x around 0 7.4%

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

   1. +-commutative 7.4%

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

   2. *-lft-identity 7.4%

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

   3. associate-*r* 7.4%

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

   4. neg-mul-1 7.4%

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

   5. distribute-rgt-out 7.4%

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

   6. unsub-neg 7.4%

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

7. Simplified 7.4%

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

8. Final simplification 7.4%

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

Alternative 16: 5.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return rem(exp(x), 1.0)
end

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. exp-neg 9.0%

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

   2. associate-*r/ 9.0%

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

   3. *-rgt-identity 9.0%

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

3. Simplified 9.0%

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

4. Taylor expanded in x around 0 8.2%

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

5. Taylor expanded in x around 0 6.7%

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

6. Final simplification 6.7%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))