Compound Interest

Percentage Accurate: 28.4% → 99.6%

Time: 17.4s

Alternatives: 20

Speedup: 38.0×

• could not determine a ground truth

  1. i = -2.0000000000000001e231
  2. n = 1.99999999999999997e77

Specification

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 28.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-53}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} + \frac{n}{i} \cdot \left(t\_0 \cdot 100\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot t\_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -2e-53)
     (+ (/ n (/ i -100.0)) (* (/ n i) (* t_0 100.0)))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (/ (* -100.0 (- n (* n t_0))) i)
         (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-53) {
		tmp = (n / (i / -100.0)) + ((n / i) * (t_0 * 100.0));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-53) {
		tmp = (n / (i / -100.0)) + ((n / i) * (t_0 * 100.0));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-53:
		tmp = (n / (i / -100.0)) + ((n / i) * (t_0 * 100.0))
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = (-100.0 * (n - (n * t_0))) / i
	else:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-53)
		tmp = Float64(Float64(n / Float64(i / -100.0)) + Float64(Float64(n / i) * Float64(t_0 * 100.0)));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-100.0 * Float64(n - Float64(n * t_0))) / i);
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-53], N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n / i), $MachinePrecision] * N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(-100.0 * N[(n - N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -2.00000000000000006e-53

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto 100 \cdot \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{100 \cdot \left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      4. associate-+l- N/A

         \[\leadsto \frac{100 \cdot \left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{100 \cdot \left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{100 \cdot \left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \]

      10. frac-2neg N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(\color{blue}{1} - {\left(1 + \frac{i}{n}\right)}^{n}\right) \]

      11. sub-neg N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}\right) \]

      12. distribute-lft-in N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} \cdot 1 + \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} + \color{blue}{\frac{-100}{\frac{i}{n}}} \cdot \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{-100}{\frac{i}{n}}\right), \color{blue}{\left(\frac{-100}{\frac{i}{n}} \cdot \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} + \frac{n}{i} \cdot \left(100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

   if -2.00000000000000006e-53 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 27.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{n}{i} \cdot -100 - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{n \cdot -100}{i} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{\frac{\frac{i}{n}}{-100}} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}}{\frac{\color{blue}{\frac{i}{n}}}{-100}} \]

      5. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{\frac{\color{blue}{i}}{n}}{-100}} \]

      6. div-inv N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n} \cdot \color{blue}{\frac{1}{-100}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{\frac{1}{-100}}} \]

      8. frac-sub N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{i \cdot \frac{1}{-100}}} \]

      9. div-inv N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\frac{i}{\color{blue}{-100}}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}\right), \color{blue}{\left(\frac{i}{-100}\right)}\right) \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{\left(n \cdot -100\right) \cdot -0.01 - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{\frac{i}{-100}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{i} \cdot \color{blue}{-100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100}{\color{blue}{i}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100\right), \color{blue}{i}\right) \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\left(n - {\left(1 + \frac{i}{n}\right)}^{n} \cdot n\right) \cdot -100}{i}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 83.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-53}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} + \frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(\frac{t\_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot t\_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-229)
     (* 100.0 (- (/ t_0 (/ i n)) (/ n i)))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (/ (* -100.0 (- n (* n t_0))) i)
         (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -1e-229:
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i))
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = (-100.0 * (n - (n * t_0))) / i
	else:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-229)
		tmp = Float64(100.0 * Float64(Float64(t_0 / Float64(i / n)) - Float64(n / i)));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-100.0 * Float64(n - Float64(n * t_0))) / i);
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-229], N[(100.0 * N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(-100.0 * N[(n - N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.00000000000000007e-229

   1. Initial program 97.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{\color{blue}{i}}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right), \color{blue}{\left(\frac{n}{i}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\frac{i}{n}\right)\right), \left(\frac{\color{blue}{n}}{i}\right)\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\frac{i}{n}\right)\right), \left(\frac{n}{i}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\frac{i}{n}\right)\right), \left(\frac{n}{i}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\frac{i}{n}\right)\right), \left(\frac{n}{i}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \mathsf{/.f64}\left(i, n\right)\right), \left(\frac{n}{i}\right)\right)\right) \]

      9. /-lowering-/.f64 97.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \mathsf{/.f64}\left(i, n\right)\right), \mathsf{/.f64}\left(n, \color{blue}{i}\right)\right)\right) \]

   4. Applied egg-rr 97.4%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)} \]

   if -1.00000000000000007e-229 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 70.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{n}{i} \cdot -100 - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{n \cdot -100}{i} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{\frac{\frac{i}{n}}{-100}} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}}{\frac{\color{blue}{\frac{i}{n}}}{-100}} \]

      5. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{\frac{\color{blue}{i}}{n}}{-100}} \]

      6. div-inv N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n} \cdot \color{blue}{\frac{1}{-100}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{\frac{1}{-100}}} \]

      8. frac-sub N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{i \cdot \frac{1}{-100}}} \]

      9. div-inv N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\frac{i}{\color{blue}{-100}}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}\right), \color{blue}{\left(\frac{i}{-100}\right)}\right) \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{\left(n \cdot -100\right) \cdot -0.01 - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{\frac{i}{-100}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{i} \cdot \color{blue}{-100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100}{\color{blue}{i}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100\right), \color{blue}{i}\right) \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\left(n - {\left(1 + \frac{i}{n}\right)}^{n} \cdot n\right) \cdot -100}{i}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 83.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t\_0 + -1\\ t_2 := \frac{t\_1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t\_1}{i}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot t\_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -1e-229)
     (* 100.0 (* n (/ t_1 i)))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_2 INFINITY)
         (/ (* -100.0 (- n (* n t_0))) i)
         (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = 100.0 * (n * (t_1 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = 100.0 * (n * (t_1 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (-100.0 * (n - (n * t_0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -1e-229:
		tmp = 100.0 * (n * (t_1 / i))
	elif t_2 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_2 <= math.inf:
		tmp = (-100.0 * (n - (n * t_0))) / i
	else:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = Float64(100.0 * Float64(n * Float64(t_1 / i)));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(-100.0 * Float64(n - Float64(n * t_0))) / i);
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], N[(100.0 * N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(-100.0 * N[(n - N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.00000000000000007e-229

   1. Initial program 97.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \color{blue}{n}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right), \color{blue}{n}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right), i\right), n\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      9. metadata-eval 97.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), -1\right), i\right), n\right)\right) \]

   4. Applied egg-rr 97.4%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]

   if -1.00000000000000007e-229 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 70.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{n}{i} \cdot -100 - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{n \cdot -100}{i} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{\frac{\frac{i}{n}}{-100}} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}}{\frac{\color{blue}{\frac{i}{n}}}{-100}} \]

      5. *-commutative N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{\frac{\color{blue}{i}}{n}}{-100}} \]

      6. div-inv N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n} \cdot \color{blue}{\frac{1}{-100}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{-100 \cdot n}{i} - \frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{\frac{1}{-100}}} \]

      8. frac-sub N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\color{blue}{i \cdot \frac{1}{-100}}} \]

      9. div-inv N/A

         \[\leadsto \frac{\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}}{\frac{i}{\color{blue}{-100}}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-100 \cdot n\right) \cdot \frac{1}{-100} - i \cdot \frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{i}{n}}\right), \color{blue}{\left(\frac{i}{-100}\right)}\right) \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{\left(n \cdot -100\right) \cdot -0.01 - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{\frac{i}{-100}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{i} \cdot \color{blue}{-100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100}{\color{blue}{i}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(n \cdot -100\right) \cdot \frac{-1}{100} - i \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \cdot -100\right), \color{blue}{i}\right) \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\left(n - {\left(1 + \frac{i}{n}\right)}^{n} \cdot n\right) \cdot -100}{i}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 83.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{-100 \cdot \left(n - n \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t\_0 + -1\\ t_2 := \frac{t\_1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t\_1}{i}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(-100 + t\_0 \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -1e-229)
     (* 100.0 (* n (/ t_1 i)))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_2 INFINITY)
         (/ (* n (+ -100.0 (* t_0 100.0))) i)
         (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = 100.0 * (n * (t_1 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (n * (-100.0 + (t_0 * 100.0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = 100.0 * (n * (t_1 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (n * (-100.0 + (t_0 * 100.0))) / i;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -1e-229:
		tmp = 100.0 * (n * (t_1 / i))
	elif t_2 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_2 <= math.inf:
		tmp = (n * (-100.0 + (t_0 * 100.0))) / i
	else:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = Float64(100.0 * Float64(n * Float64(t_1 / i)));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(n * Float64(-100.0 + Float64(t_0 * 100.0))) / i);
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], N[(100.0 * N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(n * N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.00000000000000007e-229

   1. Initial program 97.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \color{blue}{n}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right), \color{blue}{n}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right), i\right), n\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      9. metadata-eval 97.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), -1\right), i\right), n\right)\right) \]

   4. Applied egg-rr 97.4%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]

   if -1.00000000000000007e-229 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 70.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto 100 \cdot \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right)}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-100\right)\right) \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}{\mathsf{neg}\left(\frac{i}{\color{blue}{n}}\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      10. frac-2neg N/A

          \[\leadsto \frac{-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}{\color{blue}{\frac{i}{n}}} \]

      11. div-inv N/A

          \[\leadsto \left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \cdot \color{blue}{\frac{1}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \cdot \frac{n}{\color{blue}{i}} \]

      13. associate-*r/ N/A

          \[\leadsto \frac{\left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \cdot n}{\color{blue}{i}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \cdot n\right), \color{blue}{i}\right) \]

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\left(-100 + 100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot n}{i}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 83.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(n \cdot \frac{t\_0}{i}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* n (/ t_0 i)))))
   (if (<= t_1 -1e-229)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY) t_2 (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * (n * (t_0 / i));
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * (n * (t_0 / i));
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	t_2 = 100.0 * (n * (t_0 / i))
	tmp = 0
	if t_1 <= -1e-229:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(n * Float64(t_0 / i)))
	tmp = 0.0
	if (t_1 <= -1e-229)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-229], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.00000000000000007e-229 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \color{blue}{n}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right), \color{blue}{n}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right), i\right), n\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      9. metadata-eval 98.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), -1\right), i\right), n\right)\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]

   if -1.00000000000000007e-229 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 70.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 70.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 83.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ t_1 := \frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-22}:\\ \;\;\;\;100 \cdot \frac{t\_0}{i}\\ \mathbf{elif}\;n \leq -9.6 \cdot 10^{-232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 1.4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{i}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (expm1 i)))
        (t_1
         (/
          100.0
          (+ (/ 1.0 n) (* i (+ (/ -0.5 n) (/ (* i 0.08333333333333333) n)))))))
   (if (<= n -5.2e-22)
     (* 100.0 (/ t_0 i))
     (if (<= n -9.6e-232)
       t_1
       (if (<= n 2.7e-194)
         (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
         (if (<= n 1.4) t_1 (/ 100.0 (/ i t_0))))))))

C

double code(double i, double n) {
	double t_0 = n * expm1(i);
	double t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	double tmp;
	if (n <= -5.2e-22) {
		tmp = 100.0 * (t_0 / i);
	} else if (n <= -9.6e-232) {
		tmp = t_1;
	} else if (n <= 2.7e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 1.4) {
		tmp = t_1;
	} else {
		tmp = 100.0 / (i / t_0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * Math.expm1(i);
	double t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	double tmp;
	if (n <= -5.2e-22) {
		tmp = 100.0 * (t_0 / i);
	} else if (n <= -9.6e-232) {
		tmp = t_1;
	} else if (n <= 2.7e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 1.4) {
		tmp = t_1;
	} else {
		tmp = 100.0 / (i / t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * math.expm1(i)
	t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))))
	tmp = 0
	if n <= -5.2e-22:
		tmp = 100.0 * (t_0 / i)
	elif n <= -9.6e-232:
		tmp = t_1
	elif n <= 2.7e-194:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	elif n <= 1.4:
		tmp = t_1
	else:
		tmp = 100.0 / (i / t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * expm1(i))
	t_1 = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(-0.5 / n) + Float64(Float64(i * 0.08333333333333333) / n)))))
	tmp = 0.0
	if (n <= -5.2e-22)
		tmp = Float64(100.0 * Float64(t_0 / i));
	elseif (n <= -9.6e-232)
		tmp = t_1;
	elseif (n <= 2.7e-194)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	elseif (n <= 1.4)
		tmp = t_1;
	else
		tmp = Float64(100.0 / Float64(i / t_0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(i * 0.08333333333333333), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e-22], N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -9.6e-232], t$95$1, If[LessEqual[n, 2.7e-194], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4], t$95$1, N[(100.0 / N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.2e-22

   1. Initial program 31.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 87.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 87.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   if -5.2e-22 < n < -9.59999999999999995e-232 or 2.7e-194 < n < 1.3999999999999999

   1. Initial program 19.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 32.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 32.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 34.5%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{i} \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{1}{12} \cdot \frac{i}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot i}{n} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12}}{n} \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right), \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{n}\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12}}{n} \cdot i\right)\right)\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot i}{\color{blue}{n}}\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot i\right), \color{blue}{n}\right)\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

   10. Simplified 58.1%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}} \]

   if -9.59999999999999995e-232 < n < 2.7e-194

   1. Initial program 70.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 84.2%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 1.3999999999999999 < n

   1. Initial program 29.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-22}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -9.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 1.4:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ t_1 := \frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{if}\;n \leq -4 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.15 \cdot 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.122:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i)))
        (t_1
         (/
          100.0
          (+ (/ 1.0 n) (* i (+ (/ -0.5 n) (/ (* i 0.08333333333333333) n)))))))
   (if (<= n -4e-15)
     t_0
     (if (<= n -1.15e-231)
       t_1
       (if (<= n 8.5e-193)
         (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
         (if (<= n 0.122) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	double tmp;
	if (n <= -4e-15) {
		tmp = t_0;
	} else if (n <= -1.15e-231) {
		tmp = t_1;
	} else if (n <= 8.5e-193) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.122) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	double tmp;
	if (n <= -4e-15) {
		tmp = t_0;
	} else if (n <= -1.15e-231) {
		tmp = t_1;
	} else if (n <= 8.5e-193) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.122) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	t_1 = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))))
	tmp = 0
	if n <= -4e-15:
		tmp = t_0
	elif n <= -1.15e-231:
		tmp = t_1
	elif n <= 8.5e-193:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	elif n <= 0.122:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	t_1 = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(-0.5 / n) + Float64(Float64(i * 0.08333333333333333) / n)))))
	tmp = 0.0
	if (n <= -4e-15)
		tmp = t_0;
	elseif (n <= -1.15e-231)
		tmp = t_1;
	elseif (n <= 8.5e-193)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	elseif (n <= 0.122)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(i * 0.08333333333333333), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4e-15], t$95$0, If[LessEqual[n, -1.15e-231], t$95$1, If[LessEqual[n, 8.5e-193], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.122], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0000000000000003e-15 or 0.122 < n

   1. Initial program 30.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 89.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 89.6%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   if -4.0000000000000003e-15 < n < -1.15e-231 or 8.50000000000000004e-193 < n < 0.122

   1. Initial program 19.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 32.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 32.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 34.5%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{i} \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{1}{12} \cdot \frac{i}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot i}{n} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12}}{n} \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right), \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{n}\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12}}{n} \cdot i\right)\right)\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot i}{\color{blue}{n}}\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot i\right), \color{blue}{n}\right)\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

   10. Simplified 58.1%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}} \]

   if -1.15e-231 < n < 8.50000000000000004e-193

   1. Initial program 70.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 84.2%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -1.15 \cdot 10^{-231}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.122:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{1}{i}\right)\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (* (expm1 i) (/ 1.0 i)))))
   (if (<= n -1.2e-167)
     t_0
     (if (<= n 8.4e-182) (* (/ -100.0 (/ i n)) (+ 1.0 -1.0)) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1(i) * (1.0 / i));
	double tmp;
	if (n <= -1.2e-167) {
		tmp = t_0;
	} else if (n <= 8.4e-182) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1(i) * (1.0 / i));
	double tmp;
	if (n <= -1.2e-167) {
		tmp = t_0;
	} else if (n <= 8.4e-182) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1(i) * (1.0 / i))
	tmp = 0
	if n <= -1.2e-167:
		tmp = t_0
	elif n <= 8.4e-182:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(i) * Float64(1.0 / i)))
	tmp = 0.0
	if (n <= -1.2e-167)
		tmp = t_0;
	elseif (n <= 8.4e-182)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.2e-167], t$95$0, If[LessEqual[n, 8.4e-182], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.19999999999999997e-167 or 8.4000000000000001e-182 < n

   1. Initial program 26.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 75.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 75.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      3. div-inv N/A

         \[\leadsto \left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right) \cdot \color{blue}{\frac{1}{i}} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(100 \cdot n\right) \cdot \left(e^{i} - 1\right)\right) \cdot \frac{\color{blue}{1}}{i} \]

      5. associate-*l* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(\left(e^{i} - 1\right) \cdot \frac{1}{i}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\left(e^{i} - 1\right) \cdot \frac{1}{i}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{\left(e^{i} - 1\right)} \cdot \frac{1}{i}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(e^{i} - 1\right), \color{blue}{\left(\frac{1}{i}\right)}\right)\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \left(\frac{\color{blue}{1}}{i}\right)\right)\right) \]

      10. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(i\right), \left(\frac{\color{blue}{1}}{i}\right)\right)\right) \]

      11. /-lowering-/.f64 79.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(1, \color{blue}{i}\right)\right)\right) \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{1}{i}\right)} \]

   if -1.19999999999999997e-167 < n < 8.4000000000000001e-182

   1. Initial program 62.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-167}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+170}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.41:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.8e+170)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n -6.5e-166)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= n 9e-194)
       (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
       (if (<= n 0.41)
         (/
          100.0
          (+ (/ 1.0 n) (* i (+ (/ -0.5 n) (/ (* i 0.08333333333333333) n)))))
         (*
          100.0
          (/
           (*
            n
            (*
             i
             (+
              1.0
              (*
               i
               (+
                0.5
                (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
           i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.8e+170) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -6.5e-166) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 9e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.41) {
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.8e+170) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -6.5e-166) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 9e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.41) {
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.8e+170:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= -6.5e-166:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 9e-194:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	elif n <= 0.41:
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.8e+170)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= -6.5e-166)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 9e-194)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	elseif (n <= 0.41)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(-0.5 / n) + Float64(Float64(i * 0.08333333333333333) / n)))));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.8e+170], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6.5e-166], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9e-194], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.41], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(i * 0.08333333333333333), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.7999999999999998e170

   1. Initial program 7.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto n \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + -100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + \left(\mathsf{neg}\left(100\right)\right) \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} - 100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\left(\mathsf{neg}\left(100\right)\right) \cdot \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(-100 \cdot \left(\color{blue}{\frac{1}{i}} - \frac{e^{i}}{i}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\left(\frac{1}{i}\right), \color{blue}{\left(\frac{e^{i}}{i}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \left(\frac{\color{blue}{e^{i}}}{i}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\left(e^{i}\right), \color{blue}{i}\right)\right)\right)\right) \]

      14. exp-lowering-exp.f64 35.4%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(i\right), i\right)\right)\right)\right) \]

   7. Simplified 35.4%

      \[\leadsto \color{blue}{n \cdot \left(-100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(\frac{50}{3} \cdot i\right)}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(i \cdot \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   10. Simplified 70.5%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -3.7999999999999998e170 < n < -6.50000000000000019e-166

   1. Initial program 36.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 71.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if -6.50000000000000019e-166 < n < 8.9999999999999997e-194

   1. Initial program 65.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 8.9999999999999997e-194 < n < 0.409999999999999976

   1. Initial program 13.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 31.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 31.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 31.8%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{i} \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{1}{12} \cdot \frac{i}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot i}{n} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12}}{n} \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right), \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{n}\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12}}{n} \cdot i\right)\right)\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot i}{\color{blue}{n}}\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot i\right), \color{blue}{n}\right)\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

   10. Simplified 58.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}} \]

   if 0.409999999999999976 < n

   1. Initial program 29.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      8. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 87.5%

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+170}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.41:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.41:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.4e-167)
   (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n)))
   (if (<= n 3.6e-194)
     (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
     (if (<= n 0.41)
       (/
        100.0
        (+ (/ 1.0 n) (* i (+ (/ -0.5 n) (/ (* i 0.08333333333333333) n)))))
       (*
        100.0
        (/
         (*
          n
          (*
           i
           (+
            1.0
            (*
             i
             (+
              0.5
              (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
         i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-167) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 3.6e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.41) {
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.4d-167)) then
        tmp = 100.0d0 / ((1.0d0 / n) + ((i * (-0.5d0)) / n))
    else if (n <= 3.6d-194) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else if (n <= 0.41d0) then
        tmp = 100.0d0 / ((1.0d0 / n) + (i * (((-0.5d0) / n) + ((i * 0.08333333333333333d0) / n))))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-167) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 3.6e-194) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 0.41) {
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.4e-167:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	elif n <= 3.6e-194:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	elif n <= 0.41:
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.4e-167)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	elseif (n <= 3.6e-194)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	elseif (n <= 0.41)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(-0.5 / n) + Float64(Float64(i * 0.08333333333333333) / n)))));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.4e-167)
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	elseif (n <= 3.6e-194)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	elseif (n <= 0.41)
		tmp = 100.0 / ((1.0 / n) + (i * ((-0.5 / n) + ((i * 0.08333333333333333) / n))));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.4e-167], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.6e-194], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.41], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(i * 0.08333333333333333), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.4000000000000001e-167

   1. Initial program 27.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 76.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 76.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 57.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]

   if -5.4000000000000001e-167 < n < 3.6e-194

   1. Initial program 65.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 3.6e-194 < n < 0.409999999999999976

   1. Initial program 13.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 31.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 31.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 31.8%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(i \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{i} \cdot \left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{12} \cdot \frac{i}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{1}{12} \cdot \frac{i}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot i}{n} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12}}{n} \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{n}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right), \left(\left(\frac{1}{12} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{n}\right), \left(\left(\color{blue}{\frac{1}{12}} \cdot \frac{1}{n}\right) \cdot i\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\color{blue}{\left(\frac{1}{12} \cdot \frac{1}{n}\right)} \cdot i\right)\right)\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot 1}{n} \cdot i\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12}}{n} \cdot i\right)\right)\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \left(\frac{\frac{1}{12} \cdot i}{\color{blue}{n}}\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot i\right), \color{blue}{n}\right)\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{1}{12}\right), n\right)\right)\right)\right)\right) \]

   10. Simplified 58.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}} \]

   if 0.409999999999999976 < n

   1. Initial program 29.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      8. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 87.5%

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-194}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 0.41:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{-0.5}{n} + \frac{i \cdot 0.08333333333333333}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-193}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.6e-168)
   (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n)))
   (if (<= n 1.06e-193)
     (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
     (if (<= n 1.55)
       (* 100.0 (/ i (/ i n)))
       (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.6e-168) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 1.06e-193) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.6d-168)) then
        tmp = 100.0d0 / ((1.0d0 / n) + ((i * (-0.5d0)) / n))
    else if (n <= 1.06d-193) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else if (n <= 1.55d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.6e-168) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 1.06e-193) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.6e-168:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	elif n <= 1.06e-193:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	elif n <= 1.55:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.6e-168)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	elseif (n <= 1.06e-193)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	elseif (n <= 1.55)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.6e-168)
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	elseif (n <= 1.06e-193)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	elseif (n <= 1.55)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.6e-168], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.06e-193], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.60000000000000003e-168

   1. Initial program 27.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 76.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 76.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 57.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]

   if -1.60000000000000003e-168 < n < 1.06e-193

   1. Initial program 65.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 1.06e-193 < n < 1.55000000000000004

   1. Initial program 13.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 53.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.55000000000000004 < n

   1. Initial program 29.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 90.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      13. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right)\right), i\right) \]

   8. Simplified 80.4%

      \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + 50 \cdot i\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-193}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{-167}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.55e-167)
   (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n)))
   (if (<= n 4.7e-181)
     (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
     (*
      n
      (+
       100.0
       (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.55e-167) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 4.7e-181) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.55d-167)) then
        tmp = 100.0d0 / ((1.0d0 / n) + ((i * (-0.5d0)) / n))
    else if (n <= 4.7d-181) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.55e-167) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 4.7e-181) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.55e-167:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	elif n <= 4.7e-181:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.55e-167)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	elseif (n <= 4.7e-181)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.55e-167)
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	elseif (n <= 4.7e-181)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.55e-167], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.7e-181], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.55e-167

   1. Initial program 27.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 76.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 76.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 57.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]

   if -1.55e-167 < n < 4.6999999999999998e-181

   1. Initial program 62.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 4.6999999999999998e-181 < n

   1. Initial program 25.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 24.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto n \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + -100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + \left(\mathsf{neg}\left(100\right)\right) \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} - 100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\left(\mathsf{neg}\left(100\right)\right) \cdot \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(-100 \cdot \left(\color{blue}{\frac{1}{i}} - \frac{e^{i}}{i}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\left(\frac{1}{i}\right), \color{blue}{\left(\frac{e^{i}}{i}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \left(\frac{\color{blue}{e^{i}}}{i}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\left(e^{i}\right), \color{blue}{i}\right)\right)\right)\right) \]

      14. exp-lowering-exp.f64 32.6%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(i\right), i\right)\right)\right)\right) \]

   7. Simplified 32.6%

      \[\leadsto \color{blue}{n \cdot \left(-100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{50}{3} + \frac{25}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{50}{3}, \color{blue}{\left(\frac{25}{6} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{50}{3}, \left(i \cdot \color{blue}{\frac{25}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 73.3%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(i, \color{blue}{\frac{25}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 73.3%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{-167}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.4e-160)
   (/ 100.0 (+ (/ 1.0 n) (/ (* i -0.5) n)))
   (if (<= n 1.8e-182)
     (* (/ -100.0 (/ i n)) (+ 1.0 -1.0))
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-160) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 1.8e-182) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.4d-160)) then
        tmp = 100.0d0 / ((1.0d0 / n) + ((i * (-0.5d0)) / n))
    else if (n <= 1.8d-182) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-160) {
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	} else if (n <= 1.8e-182) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.4e-160:
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n))
	elif n <= 1.8e-182:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.4e-160)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(Float64(i * -0.5) / n)));
	elseif (n <= 1.8e-182)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.4e-160)
		tmp = 100.0 / ((1.0 / n) + ((i * -0.5) / n));
	elseif (n <= 1.8e-182)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.4e-160], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-182], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.40000000000000008e-160

   1. Initial program 27.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 76.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 76.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{i}{n \cdot \left(e^{i} - 1\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(e^{i} - 1\right)}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right)\right) \]

   7. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n} + \frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{1}{n} + \color{blue}{\frac{-1}{2} \cdot \frac{i}{n}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{i}{n}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{i}{n}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{-1}{2} \cdot i}{\color{blue}{n}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{n}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(i \cdot \frac{-1}{2}\right), n\right)\right)\right) \]

      7. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), n\right)\right)\right) \]

   10. Simplified 57.2%

       \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}} \]

   if -1.40000000000000008e-160 < n < 1.79999999999999988e-182

   1. Initial program 62.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]

   if 1.79999999999999988e-182 < n

   1. Initial program 25.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 24.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto n \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + -100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + \left(\mathsf{neg}\left(100\right)\right) \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} - 100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\left(\mathsf{neg}\left(100\right)\right) \cdot \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(-100 \cdot \left(\color{blue}{\frac{1}{i}} - \frac{e^{i}}{i}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\left(\frac{1}{i}\right), \color{blue}{\left(\frac{e^{i}}{i}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \left(\frac{\color{blue}{e^{i}}}{i}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\left(e^{i}\right), \color{blue}{i}\right)\right)\right)\right) \]

      14. exp-lowering-exp.f64 32.6%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(i\right), i\right)\right)\right)\right) \]

   7. Simplified 32.6%

      \[\leadsto \color{blue}{n \cdot \left(-100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(\frac{50}{3} \cdot i\right)}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(i \cdot \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.75 \cdot 10^{-180}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -1.45e-155)
     t_0
     (if (<= n 3.75e-180) (* (/ -100.0 (/ i n)) (+ 1.0 -1.0)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -1.45e-155) {
		tmp = t_0;
	} else if (n <= 3.75e-180) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    if (n <= (-1.45d-155)) then
        tmp = t_0
    else if (n <= 3.75d-180) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -1.45e-155) {
		tmp = t_0;
	} else if (n <= 3.75e-180) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	tmp = 0
	if n <= -1.45e-155:
		tmp = t_0
	elif n <= 3.75e-180:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -1.45e-155)
		tmp = t_0;
	elseif (n <= 3.75e-180)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	tmp = 0.0;
	if (n <= -1.45e-155)
		tmp = t_0;
	elseif (n <= 3.75e-180)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.45e-155], t$95$0, If[LessEqual[n, 3.75e-180], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.45000000000000005e-155 or 3.75000000000000008e-180 < n

   1. Initial program 26.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{i}{n}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{\mathsf{neg}\left(-100\right)}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      11. frac-2neg N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\color{blue}{\frac{i}{n}}} \]

      12. clear-num N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      13. un-div-inv N/A

          \[\leadsto \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{n}}{-100}}} \]

      14. div-sub N/A

          \[\leadsto \frac{1}{\frac{\frac{i}{n}}{-100}} - \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

      15. clear-num N/A

          \[\leadsto \frac{-100}{\frac{i}{n}} - \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{\frac{i}{n}}{-100}} \]

   4. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{n}}{-100}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto n \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(\left(-100 \cdot \frac{e^{i}}{i} + 100 \cdot \frac{1}{i}\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + -100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} + \left(\mathsf{neg}\left(100\right)\right) \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\left(100 \cdot \frac{1}{i} - 100 \cdot \frac{e^{i}}{i}\right)\right)\right)\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(\left(\mathsf{neg}\left(100\right)\right) \cdot \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(-100 \cdot \left(\color{blue}{\frac{1}{i}} - \frac{e^{i}}{i}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \color{blue}{\left(\frac{1}{i} - \frac{e^{i}}{i}\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\left(\frac{1}{i}\right), \color{blue}{\left(\frac{e^{i}}{i}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \left(\frac{\color{blue}{e^{i}}}{i}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\left(e^{i}\right), \color{blue}{i}\right)\right)\right)\right) \]

      14. exp-lowering-exp.f64 33.2%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-100, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, i\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(i\right), i\right)\right)\right)\right) \]

   7. Simplified 33.2%

      \[\leadsto \color{blue}{n \cdot \left(-100 \cdot \left(\frac{1}{i} - \frac{e^{i}}{i}\right)\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(\frac{50}{3} \cdot i\right)}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(i \cdot \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 62.9%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   10. Simplified 62.9%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -1.45000000000000005e-155 < n < 3.75000000000000008e-180

   1. Initial program 62.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.75 \cdot 10^{-180}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.26 \cdot 10^{-181}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -6.2e-167)
     t_0
     (if (<= n 2.26e-181) (* (/ -100.0 (/ i n)) (+ 1.0 -1.0)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -6.2e-167) {
		tmp = t_0;
	} else if (n <= 2.26e-181) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-6.2d-167)) then
        tmp = t_0
    else if (n <= 2.26d-181) then
        tmp = ((-100.0d0) / (i / n)) * (1.0d0 + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -6.2e-167) {
		tmp = t_0;
	} else if (n <= 2.26e-181) {
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -6.2e-167:
		tmp = t_0
	elif n <= 2.26e-181:
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -6.2e-167)
		tmp = t_0;
	elseif (n <= 2.26e-181)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -6.2e-167)
		tmp = t_0;
	elseif (n <= 2.26e-181)
		tmp = (-100.0 / (i / n)) * (1.0 + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-167], t$95$0, If[LessEqual[n, 2.26e-181], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.2e-167 or 2.25999999999999993e-181 < n

   1. Initial program 26.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 75.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 75.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot n + \color{blue}{50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* N/A

         \[\leadsto 100 \cdot n + \left(50 \cdot i\right) \cdot \color{blue}{n} \]

      3. distribute-rgt-out N/A

         \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 58.1%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 58.1%

      \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   if -6.2e-167 < n < 2.25999999999999993e-181

   1. Initial program 62.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot \color{blue}{100} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\color{blue}{\frac{i}{n}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100\right)}{\frac{\color{blue}{i}}{n}} \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\frac{i}{n}}\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100}{\color{blue}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      11. associate-*r/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \color{blue}{\frac{100}{\mathsf{neg}\left(\frac{i}{n}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{1 \cdot 100}{\mathsf{neg}\left(\color{blue}{\frac{i}{n}}\right)} \]

      13. associate-*l/ N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot \color{blue}{100}\right) \]

      14. distribute-neg-frac2 N/A

          \[\leadsto \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right) \cdot \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{1}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \frac{-100}{\frac{i}{n}} \cdot \left(1 - \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-167}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 2.26 \cdot 10^{-181}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+45}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+45)
   (* 100.0 (/ (* i n) i))
   (if (<= n 1.55) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+45) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.6d+45)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 1.55d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+45) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.6e+45:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 1.55:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e+45)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 1.55)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.6e+45)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 1.55)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e+45], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e45

   1. Initial program 30.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 87.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 87.9%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot n\right)}, i\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot i\right), i\right)\right) \]

      2. *-lowering-*.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, i\right), i\right)\right) \]

   8. Simplified 52.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot i}}{i} \]

   if -3.6e45 < n < 1.55000000000000004

   1. Initial program 37.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 53.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.55000000000000004 < n

   1. Initial program 29.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot n + \color{blue}{50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* N/A

         \[\leadsto 100 \cdot n + \left(50 \cdot i\right) \cdot \color{blue}{n} \]

      3. distribute-rgt-out N/A

         \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 69.1%

      \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+45}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -1.6 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-34}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -1.6e+45) t_0 (if (<= n 4e-34) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -1.6e+45) {
		tmp = t_0;
	} else if (n <= 4e-34) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * ((i * n) / i)
    if (n <= (-1.6d+45)) then
        tmp = t_0
    else if (n <= 4d-34) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -1.6e+45) {
		tmp = t_0;
	} else if (n <= 4e-34) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	tmp = 0
	if n <= -1.6e+45:
		tmp = t_0
	elif n <= 4e-34:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	tmp = 0.0
	if (n <= -1.6e+45)
		tmp = t_0;
	elseif (n <= 4e-34)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	tmp = 0.0;
	if (n <= -1.6e+45)
		tmp = t_0;
	elseif (n <= 4e-34)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.6e+45], t$95$0, If[LessEqual[n, 4e-34], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.6000000000000001e45 or 3.99999999999999971e-34 < n

   1. Initial program 29.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 87.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 87.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot n\right)}, i\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot i\right), i\right)\right) \]

      2. *-lowering-*.f64 59.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, i\right), i\right)\right) \]

   8. Simplified 59.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot i}}{i} \]

   if -1.6000000000000001e45 < n < 3.99999999999999971e-34

   1. Initial program 39.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 52.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+45}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-34}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 10600000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5e+37)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 10600000000.0) (* n 100.0) (* n (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5e+37) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 10600000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5d+37)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 10600000000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5e+37) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 10600000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5e+37:
		tmp = 100.0 * (i / (i / n))
	elif i <= 10600000000.0:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5e+37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 10600000000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5e+37)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 10600000000.0)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5e+37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 10600000000.0], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.99999999999999989e37

   1. Initial program 72.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 28.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -4.99999999999999989e37 < i < 1.06e10

   1. Initial program 13.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{100 \cdot n} \]

   if 1.06e10 < i

   1. Initial program 52.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 55.0%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + \frac{1}{2} \cdot i\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot i\right)\right)\right)\right), i\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

      4. *-lowering-*.f64 43.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot 0.5\right)\right)}}{i} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot \color{blue}{n} \]

       2. *-commutative N/A

          \[\leadsto n \cdot \color{blue}{\left(50 \cdot i\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(50 \cdot i\right)}\right) \]

       4. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right) \]

   11. Simplified 23.9%

       \[\leadsto \color{blue}{n \cdot \left(50 \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 10600000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 54.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 175000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 175000000000.0) (* n 100.0) (* n (* i 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 175000000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 175000000000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 175000000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 175000000000.0:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 175000000000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 175000000000.0)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 175000000000.0], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.75e11

   1. Initial program 27.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 57.4%

      \[\leadsto \color{blue}{100 \cdot n} \]

   if 1.75e11 < i

   1. Initial program 52.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 55.0%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + \frac{1}{2} \cdot i\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot i\right)\right)\right)\right), i\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

      4. *-lowering-*.f64 43.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 43.9%

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot 0.5\right)\right)}}{i} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot \color{blue}{n} \]

       2. *-commutative N/A

          \[\leadsto n \cdot \color{blue}{\left(50 \cdot i\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(50 \cdot i\right)}\right) \]

       4. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right) \]

   11. Simplified 23.9%

       \[\leadsto \color{blue}{n \cdot \left(50 \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 175000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 49.0% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 33.2%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 45.3%

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

5. Simplified 45.3%

   \[\leadsto \color{blue}{100 \cdot n} \]

6. Final simplification 45.3%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer Target 1: 34.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024155 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))