expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 63.4%

Time: 19.2s

Alternatives: 4

Speedup: 5.0×

• could not determine a ground truth

  1. x = -6.593308403182123e+194

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: 63.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (exp (- x))))
   (if (<= (* (fmod (exp x) t_0) t_1) 2.0)
     (/ (fmod (exp x) (* 3.0 (log (cbrt (exp t_0))))) (exp x))
     t_1)))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = exp(-x);
	double tmp;
	if ((fmod(exp(x), t_0) * t_1) <= 2.0) {
		tmp = fmod(exp(x), (3.0 * log(cbrt(exp(t_0))))) / exp(x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), t_0) * t_1) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(3.0 * log(cbrt(exp(t_0))))) / exp(x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

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))) < 2

   1. Initial program 9.4%

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

      1. /-rgt-identity 9.4%

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

      2. associate-/r/ 9.4%

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

      3. exp-neg 9.5%

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

      4. remove-double-neg 9.5%

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

   3. Simplified 9.5%

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

      1. add-log-exp 9.5%

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

      2. add-cube-cbrt 55.2%

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

      3. log-prod 55.3%

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

      4. pow2 55.3%

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

   6. Applied egg-rr 55.3%

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

      1. log-pow 55.3%

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

      2. distribute-lft1-in 55.3%

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

      3. metadata-eval 55.3%

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

   8. Simplified 55.3%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow1/3 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow3 0.0%

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

      5. log-pow 0.0%

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

      6. log-div 0.0%

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

      7. add-log-exp 0.1%

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

   6. Applied egg-rr 0.1%

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

   7. Taylor expanded in x around inf 96.3%

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

      1. neg-mul-1 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 63.8%

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

Alternative 2: 62.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-16) (exp (- (log (fmod (exp x) 1.0)) x)) (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-16) {
		tmp = exp((log(fmod(exp(x), 1.0)) - x));
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-16)) then
        tmp = exp((log(mod(exp(x), 1.0d0)) - x))
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-16:
		tmp = math.exp((math.log(math.fmod(math.exp(x), 1.0)) - x))
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-16)
		tmp = exp(Float64(log(rem(exp(x), 1.0)) - x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-16], N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999998e-17

   1. Initial program 77.3%

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

      1. /-rgt-identity 77.3%

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

      2. associate-/r/ 77.1%

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

      3. exp-neg 77.8%

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

      4. remove-double-neg 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in x around 0 77.8%

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

      1. add-exp-log 77.8%

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

      2. div-exp 78.5%

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

   7. Applied egg-rr 78.5%

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

   if -9.9999999999999998e-17 < x

   1. Initial program 5.0%

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

      1. /-rgt-identity 5.0%

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

      2. associate-/r/ 5.0%

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

      3. exp-neg 5.0%

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

      4. remove-double-neg 5.0%

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

   3. Simplified 5.0%

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

      1. add-cbrt-cube 5.0%

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

      2. pow1/3 4.9%

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

      3. pow-to-exp 4.9%

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

      4. pow3 4.9%

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

      5. log-pow 4.9%

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

      6. log-div 4.9%

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

      7. add-log-exp 4.9%

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

   6. Applied egg-rr 4.9%

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

   7. Taylor expanded in x around inf 62.1%

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

      1. neg-mul-1 62.1%

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

   9. Simplified 62.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-16) (/ (fmod (exp x) 1.0) (exp x)) (exp (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-16) {
		tmp = fmod(exp(x), 1.0) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-16)) then
        tmp = mod(exp(x), 1.0d0) / exp(x)
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= -1e-16:
		tmp = math.fmod(math.exp(x), 1.0) / math.exp(x)
	else:
		tmp = math.exp(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-16)
		tmp = Float64(rem(exp(x), 1.0) / exp(x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-16], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999998e-17

   1. Initial program 77.3%

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

      1. /-rgt-identity 77.3%

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

      2. associate-/r/ 77.1%

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

      3. exp-neg 77.8%

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

      4. remove-double-neg 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in x around 0 77.8%

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

   if -9.9999999999999998e-17 < x

   1. Initial program 5.0%

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

      1. /-rgt-identity 5.0%

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

      2. associate-/r/ 5.0%

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

      3. exp-neg 5.0%

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

      4. remove-double-neg 5.0%

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

   3. Simplified 5.0%

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

      1. add-cbrt-cube 5.0%

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

      2. pow1/3 4.9%

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

      3. pow-to-exp 4.9%

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

      4. pow3 4.9%

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

      5. log-pow 4.9%

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

      6. log-div 4.9%

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

      7. add-log-exp 4.9%

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

   6. Applied egg-rr 4.9%

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

   7. Taylor expanded in x around inf 62.1%

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

      1. neg-mul-1 62.1%

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

   9. Simplified 62.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. /-rgt-identity 7.5%

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

   2. associate-/r/ 7.5%

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

   3. exp-neg 7.5%

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

   4. remove-double-neg 7.5%

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

3. Simplified 7.5%

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

   1. add-cbrt-cube 7.5%

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

   2. pow1/3 7.5%

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

   3. pow-to-exp 7.5%

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

   4. pow3 7.5%

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

   5. log-pow 7.5%

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

   6. log-div 7.5%

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

   7. add-log-exp 7.5%

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

6. Applied egg-rr 7.5%

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

7. Taylor expanded in x around inf 61.0%

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

   1. neg-mul-1 61.0%

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

9. Simplified 61.0%

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

10. Final simplification 61.0%

    \[\leadsto e^{-x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))