expfmod (used to be hard to sample)

Percentage Accurate: 7.2% → 43.6%

Time: 16.5s

Alternatives: 5

Speedup: 5.0×

• could not determine a ground truth

  1. x = -4.227723450979604e+214

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.2% 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: 43.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.0)
   (exp x)
   (/ (fmod (exp x) (cbrt (pow (cos x) 1.5))) (exp x))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0) {
		tmp = exp(x);
	} else {
		tmp = fmod(exp(x), cbrt(pow(cos(x), 1.5))) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.0)
		tmp = exp(x);
	else
		tmp = Float64(rem(exp(x), cbrt((cos(x) ^ 1.5))) / exp(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 0.0], N[Exp[x], $MachinePrecision], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[Power[N[Cos[x], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.4%

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

      1. exp-neg 4.4%

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

      2. div-inv 4.4%

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

      3. add-cube-cbrt 4.4%

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

      4. pow3 4.4%

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

      5. pow-to-exp 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around inf 49.4%

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

      1. *-commutative 49.4%

         \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

   7. Simplified 49.4%

      \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

   8. Taylor expanded in x around 0 49.4%

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

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

   1. Initial program 14.3%

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

      1. /-rgt-identity 14.3%

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

      2. associate-/r/ 14.3%

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

      3. exp-neg 14.3%

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

      4. remove-double-neg 14.3%

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

   3. Simplified 14.3%

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

      1. add-cbrt-cube 14.3%

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

      2. pow3 14.4%

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

      3. pow1/2 14.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\color{blue}{\left({\cos x}^{0.5}\right)}}^{3}}\right)\right)}{e^{x}} \]

      4. pow-pow 14.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{\color{blue}{{\cos x}^{\left(0.5 \cdot 3\right)}}}\right)\right)}{e^{x}} \]

      5. metadata-eval 14.4%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{\color{blue}{1.5}}}\right)\right)}{e^{x}} \]

   6. Applied egg-rr 14.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{{\cos x}^{1.5}}\right)}\right)}{e^{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\sqrt{\cos x}}}\\ \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({t\_0}^{2}\right) + \log t\_0\right)\right)}{e^{x}} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. /-rgt-identity 6.8%

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

   2. associate-/r/ 6.8%

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

   3. exp-neg 6.8%

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

   4. remove-double-neg 6.8%

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

3. Simplified 6.8%

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

   1. add-log-exp 6.8%

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

   2. add-cube-cbrt 40.6%

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

   3. log-prod 40.6%

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

   4. pow2 40.6%

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

6. Applied egg-rr 40.6%

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

7. Final simplification 40.6%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
8. Add Preprocessing

Alternative 3: 43.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* t_0 (exp (- x))) 0.0) (exp x) (/ 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 = exp(x);
	} 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 = exp(x)
    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.exp(x)
	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 = exp(x);
	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[Exp[x], $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 4.4%

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

      1. exp-neg 4.4%

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

      2. div-inv 4.4%

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

      3. add-cube-cbrt 4.4%

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

      4. pow3 4.4%

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

      5. pow-to-exp 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around inf 49.4%

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

      1. *-commutative 49.4%

         \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

   7. Simplified 49.4%

      \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

   8. Taylor expanded in x around 0 49.4%

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

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

   1. Initial program 14.3%

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

      1. /-rgt-identity 14.3%

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

      2. associate-/r/ 14.3%

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

      3. exp-neg 14.3%

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

      4. remove-double-neg 14.3%

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

   3. Simplified 14.3%

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

4. Final simplification 40.7%

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

Alternative 4: 41.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ e^{\left|x\right|} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return exp(abs(x))
end

MATLAB

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

Wolfram

code[x_] := N[Exp[N[Abs[x], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. div-inv 6.8%

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

   3. add-cube-cbrt 6.8%

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

   4. pow3 6.8%

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

   5. pow-to-exp 6.8%

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

4. Applied egg-rr 5.1%

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

5. Taylor expanded in x around inf 38.9%

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

   1. *-commutative 38.9%

      \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

7. Simplified 38.9%

   \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]
8. Step-by-step derivation

   1. associate-*l* 38.9%

      \[\leadsto e^{\color{blue}{x \cdot \left(0.3333333333333333 \cdot 3\right)}} \]

   2. metadata-eval 38.9%

      \[\leadsto e^{x \cdot \color{blue}{1}} \]

   3. *-rgt-identity 38.9%

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

   4. add-sqr-sqrt 2.6%

      \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   5. sqrt-prod 38.9%

      \[\leadsto e^{\color{blue}{\sqrt{x \cdot x}}} \]

   6. rem-sqrt-square 38.9%

      \[\leadsto e^{\color{blue}{\left|x\right|}} \]

9. Applied egg-rr 38.9%

   \[\leadsto e^{\color{blue}{\left|x\right|}} \]

10. Final simplification 38.9%

    \[\leadsto e^{\left|x\right|} \]
11. Add Preprocessing

Alternative 5: 41.6% 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(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. exp-neg 6.8%

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

   2. div-inv 6.8%

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

   3. add-cube-cbrt 6.8%

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

   4. pow3 6.8%

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

   5. pow-to-exp 6.8%

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

4. Applied egg-rr 5.1%

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

5. Taylor expanded in x around inf 38.9%

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

   1. *-commutative 38.9%

      \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

7. Simplified 38.9%

   \[\leadsto e^{\color{blue}{\left(x \cdot 0.3333333333333333\right)} \cdot 3} \]

8. Taylor expanded in x around 0 38.9%

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

9. Final simplification 38.9%

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

Reproduce

herbie shell --seed 2024067 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))