expfmod (used to be hard to sample)

Percentage Accurate: 6.7% → 62.3%

Time: 20.4s

Alternatives: 3

Speedup: 5.0×

• could not determine a ground truth

  1. x = -519317929686.2718

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: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\sqrt{\cos x}}}\\ \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\log \left({t\_0}^{2}\right) + \log t\_0\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (sqrt (cos x))))))
   (if (<= x 0.002)
     (/ (fmod (exp x) (+ (log (pow t_0 2.0)) (log t_0))) (exp x))
     (exp (- x)))))

C

double code(double x) {
	double t_0 = cbrt(exp(sqrt(cos(x))));
	double tmp;
	if (x <= 0.002) {
		tmp = fmod(exp(x), (log(pow(t_0, 2.0)) + log(t_0))) / exp(x);
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(exp(sqrt(cos(x))))
	tmp = 0.0
	if (x <= 0.002)
		tmp = Float64(rem(exp(x), Float64(log((t_0 ^ 2.0)) + log(t_0))) / exp(x));
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 0.002], 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], N[Exp[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 7.8%

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

      1. /-rgt-identity 7.8%

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

      2. associate-/r/ 7.8%

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

      3. exp-neg 7.8%

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

      4. remove-double-neg 7.8%

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

   3. Simplified 7.8%

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

      1. add-log-exp 7.8%

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

      2. add-cube-cbrt 50.5%

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

      3. log-prod 50.5%

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

      4. pow2 50.5%

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

   6. Applied egg-rr 50.5%

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

   if 2e-3 < 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-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow1/3 0.0%

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

      5. log-pow 0.0%

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

      6. log-div 0.0%

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

      7. add-log-exp 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

Alternative 2: 60.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(((1.0 - x) + -1.0));
}

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = exp(((1.0 - x) + -1.0));
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

   3. pow-to-exp 6.5%

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

   4. pow1/3 6.5%

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

   5. log-pow 6.5%

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

   6. log-div 6.5%

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

   7. add-log-exp 6.5%

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

6. Applied egg-rr 6.5%

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

7. Taylor expanded in x around inf 57.4%

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

   1. *-commutative 57.4%

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

9. Simplified 57.4%

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

    1. expm1-log1p-u 40.2%

       \[\leadsto e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot -0.3333333333333333\right) \cdot 3\right)\right)}} \]

    2. expm1-undefine 40.2%

       \[\leadsto e^{\color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot -0.3333333333333333\right) \cdot 3\right)} - 1}} \]

    3. *-commutative 40.2%

       \[\leadsto e^{e^{\mathsf{log1p}\left(\color{blue}{3 \cdot \left(x \cdot -0.3333333333333333\right)}\right)} - 1} \]

    4. *-commutative 40.2%

       \[\leadsto e^{e^{\mathsf{log1p}\left(3 \cdot \color{blue}{\left(-0.3333333333333333 \cdot x\right)}\right)} - 1} \]

    5. associate-*r* 40.2%

       \[\leadsto e^{e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot -0.3333333333333333\right) \cdot x}\right)} - 1} \]

    6. metadata-eval 40.2%

       \[\leadsto e^{e^{\mathsf{log1p}\left(\color{blue}{-1} \cdot x\right)} - 1} \]

    7. neg-mul-1 40.2%

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

    8. log1p-define 40.2%

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

    9. sub-neg 40.2%

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

    10. add-exp-log 57.4%

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

11. Applied egg-rr 57.4%

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

12. Final simplification 57.4%

    \[\leadsto e^{\left(1 - x\right) + -1} \]
13. Add Preprocessing

Alternative 3: 60.7% 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 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

   3. pow-to-exp 6.5%

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

   4. pow1/3 6.5%

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

   5. log-pow 6.5%

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

   6. log-div 6.5%

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

   7. add-log-exp 6.5%

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

6. Applied egg-rr 6.5%

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

7. Taylor expanded in x around inf 57.4%

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

   1. *-commutative 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in x around 0 57.4%

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

    1. neg-mul-1 57.4%

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

12. Simplified 57.4%

    \[\leadsto e^{\color{blue}{-x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024098 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))