expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 43.3%

Time: 20.9s

Alternatives: 13

Speedup: 1.2×

• could not determine a ground truth

  1. x = -3.220010810799578e+195

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({\left(\sqrt{2}\right)}^{2} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* t_0 (exp (- x))) 0.0)
     (fmod (exp x) (+ (pow (sqrt 2.0) 2.0) -1.0))
     (/ 1.0 (/ (exp x) t_0)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((t_0 * exp(-x)) <= 0.0) {
		tmp = fmod(exp(x), (pow(sqrt(2.0), 2.0) + -1.0));
	} else {
		tmp = 1.0 / (exp(x) / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    if ((t_0 * exp(-x)) <= 0.0d0) then
        tmp = mod(exp(x), ((sqrt(2.0d0) ** 2.0d0) + (-1.0d0)))
    else
        tmp = 1.0d0 / (exp(x) / t_0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (t_0 * math.exp(-x)) <= 0.0:
		tmp = math.fmod(math.exp(x), (math.pow(math.sqrt(2.0), 2.0) + -1.0))
	else:
		tmp = 1.0 / (math.exp(x) / t_0)
	return tmp

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.2%

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

      1. exp-neg 4.2%

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

      2. associate-*r/ 4.2%

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

      3. *-rgt-identity 4.2%

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

   3. Simplified 4.2%

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

      1. expm1-log1p-u 4.2%

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

      2. expm1-udef 4.2%

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

      3. log1p-udef 4.2%

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

      4. add-exp-log 4.2%

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

   5. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. add-sqr-sqrt 49.4%

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

      4. hypot-1-def 49.4%

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

      5. pow1/2 49.4%

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

      6. sqrt-pow1 49.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, \color{blue}{{\cos x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

      7. metadata-eval 49.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{\color{blue}{0.25}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

   7. Applied egg-rr 49.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2}} - 1\right)\right)}{e^{x}} \]

   8. Taylor expanded in x around 0 49.4%

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

   9. Taylor expanded in x around 0 49.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 23.6%

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

      1. exp-neg 23.7%

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

      2. associate-*r/ 23.8%

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

      3. *-rgt-identity 23.8%

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

   3. Simplified 23.8%

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

      1. add-cbrt-cube 23.8%

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

      2. pow1/3 23.8%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

      3. pow3 23.8%

         \[\leadsto {\color{blue}{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}\right)}}^{0.3333333333333333} \]

   5. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 23.8%

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

      2. rem-cbrt-cube 23.8%

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

      3. clear-num 23.8%

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

   7. Applied egg-rr 23.8%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({\left(\sqrt{2}\right)}^{2} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\\ \end{array} \]

Alternative 2: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({\left(\sqrt{2}\right)}^{2} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* t_0 (exp (- x))) 0.0)
     (fmod (exp x) (+ (pow (sqrt 2.0) 2.0) -1.0))
     (/ t_0 (exp x)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((t_0 * exp(-x)) <= 0.0) {
		tmp = fmod(exp(x), (pow(sqrt(2.0), 2.0) + -1.0));
	} else {
		tmp = t_0 / exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    if ((t_0 * exp(-x)) <= 0.0d0) then
        tmp = mod(exp(x), ((sqrt(2.0d0) ** 2.0d0) + (-1.0d0)))
    else
        tmp = t_0 / exp(x)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (t_0 * math.exp(-x)) <= 0.0:
		tmp = math.fmod(math.exp(x), (math.pow(math.sqrt(2.0), 2.0) + -1.0))
	else:
		tmp = t_0 / math.exp(x)
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.2%

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

      1. exp-neg 4.2%

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

      2. associate-*r/ 4.2%

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

      3. *-rgt-identity 4.2%

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

   3. Simplified 4.2%

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

      1. expm1-log1p-u 4.2%

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

      2. expm1-udef 4.2%

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

      3. log1p-udef 4.2%

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

      4. add-exp-log 4.2%

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

   5. Applied egg-rr 4.2%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. add-sqr-sqrt 49.4%

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

      4. hypot-1-def 49.4%

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

      5. pow1/2 49.4%

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

      6. sqrt-pow1 49.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, \color{blue}{{\cos x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

      7. metadata-eval 49.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{\color{blue}{0.25}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

   7. Applied egg-rr 49.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2}} - 1\right)\right)}{e^{x}} \]

   8. Taylor expanded in x around 0 49.4%

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

   9. Taylor expanded in x around 0 49.4%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 23.6%

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

      1. exp-neg 23.7%

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

      2. associate-*r/ 23.8%

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

      3. *-rgt-identity 23.8%

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

   3. Simplified 23.8%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left({\left(\sqrt{2}\right)}^{2} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \]

Alternative 3: 43.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2} + -1\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ (pow (hypot 1.0 (pow (cos x) 0.25)) 2.0) -1.0)) (exp x)))

C

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Power[N[Sqrt[1.0 ^ 2 + N[Power[N[Cos[x], $MachinePrecision], 0.25], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

   1. add-sqr-sqrt 43.8%

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

   2. pow2 43.8%

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

   3. add-sqr-sqrt 43.8%

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

   4. hypot-1-def 43.8%

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

   5. pow1/2 43.8%

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

   6. sqrt-pow1 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, \color{blue}{{\cos x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

   7. metadata-eval 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{\color{blue}{0.25}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

7. Applied egg-rr 43.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2}} - 1\right)\right)}{e^{x}} \]

8. Final simplification 43.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2} + -1\right)\right)}{e^{x}} \]

Alternative 4: 42.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ (pow (sqrt 2.0) 2.0) -1.0)) (exp x)))

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

   1. add-sqr-sqrt 43.8%

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

   2. pow2 43.8%

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

   3. add-sqr-sqrt 43.8%

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

   4. hypot-1-def 43.8%

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

   5. pow1/2 43.8%

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

   6. sqrt-pow1 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, \color{blue}{{\cos x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

   7. metadata-eval 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{\color{blue}{0.25}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

7. Applied egg-rr 43.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2}} - 1\right)\right)}{e^{x}} \]

8. Taylor expanded in x around 0 43.6%

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

9. Final simplification 43.6%

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

Alternative 5: 41.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

   1. add-sqr-sqrt 43.8%

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

   2. pow2 43.8%

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

   3. add-sqr-sqrt 43.8%

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

   4. hypot-1-def 43.8%

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

   5. pow1/2 43.8%

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

   6. sqrt-pow1 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, \color{blue}{{\cos x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

   7. metadata-eval 43.8%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({\left(\mathsf{hypot}\left(1, {\cos x}^{\color{blue}{0.25}}\right)\right)}^{2} - 1\right)\right)}{e^{x}} \]

7. Applied egg-rr 43.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{{\left(\mathsf{hypot}\left(1, {\cos x}^{0.25}\right)\right)}^{2}} - 1\right)\right)}{e^{x}} \]

8. Taylor expanded in x around 0 43.6%

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

9. Taylor expanded in x around 0 42.0%

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

10. Final simplification 42.0%

    \[\leadsto \left(\left(e^{x}\right) \bmod \left({\left(\sqrt{2}\right)}^{2} + -1\right)\right) \]

Alternative 6: 6.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

6. Taylor expanded in x around 0 8.3%

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

   1. *-commutative 8.2%

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

   2. unpow2 8.2%

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

8. Simplified 8.3%

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

9. Final simplification 8.3%

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

Alternative 7: 6.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

6. Taylor expanded in x around 0 8.2%

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

   1. *-commutative 8.2%

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

   2. unpow2 8.2%

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

8. Simplified 8.2%

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

9. Final simplification 8.2%

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

Alternative 8: 6.5% 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 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in x around 0 8.2%

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

   1. *-commutative 8.2%

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

   2. unpow2 8.2%

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

6. Simplified 8.2%

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

7. Final simplification 8.2%

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

Alternative 9: 6.3% 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 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. add-exp-log 8.3%

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

5. Applied egg-rr 8.3%

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

   1. associate--l+ 8.3%

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

7. Simplified 8.3%

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

8. Taylor expanded in x around 0 8.0%

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

9. Final simplification 8.0%

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

Alternative 10: 6.3% accurate, 1.6× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

   1. add-cbrt-cube 8.3%

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

   2. pow1/3 8.3%

      \[\leadsto \color{blue}{{\left(\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{0.3333333333333333}} \]

   3. pow3 8.3%

      \[\leadsto {\color{blue}{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}\right)}}^{0.3333333333333333} \]

5. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 8.3%

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

   2. rem-cbrt-cube 8.3%

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

   3. expm1-log1p-u 8.3%

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

   4. expm1-udef 8.3%

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

   5. log1p-udef 8.3%

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

   6. +-commutative 8.3%

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

   7. add-exp-log 8.3%

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

7. Applied egg-rr 8.3%

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

8. Taylor expanded in x around 0 8.0%

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

9. Final simplification 8.0%

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

Alternative 11: 6.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in x around 0 8.0%

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

   1. div-inv 8.0%

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

   2. rec-exp 7.9%

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

6. Applied egg-rr 7.9%

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

   1. exp-neg 8.0%

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

   2. div-inv 8.0%

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

   3. clear-num 8.0%

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

8. Applied egg-rr 8.0%

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

9. Final simplification 8.0%

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

Alternative 12: 6.3% 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 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in x around 0 8.0%

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

5. Final simplification 8.0%

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

Alternative 13: 5.3% 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 8.3%

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

   1. exp-neg 8.3%

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

   2. associate-*r/ 8.3%

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

   3. *-rgt-identity 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in x around 0 8.0%

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

5. Taylor expanded in x around 0 6.4%

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

6. Final simplification 6.4%

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

Reproduce

herbie shell --seed 2023203 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))