Compound Interest

Percentage Accurate: 27.9% → 95.9%

Time: 18.8s

Alternatives: 22

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: 27.9% 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: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 - t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \frac{\frac{t\_0}{\frac{1}{n}} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-61)
     (* (/ -100.0 (/ i n)) (- 1.0 t_0))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (/ (- (/ t_0 (/ 1.0 n)) n) i))
         (* n 100.0))))))

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-61) {
		tmp = (-100.0 / (i / n)) * (1.0 - t_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 * (((t_0 / (1.0 / n)) - n) / i);
	} else {
		tmp = n * 100.0;
	}
	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-61) {
		tmp = (-100.0 / (i / n)) * (1.0 - t_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 * (((t_0 / (1.0 / n)) - n) / i);
	} else {
		tmp = n * 100.0;
	}
	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-61:
		tmp = (-100.0 / (i / n)) * (1.0 - t_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 * (((t_0 / (1.0 / n)) - n) / i)
	else:
		tmp = n * 100.0
	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-61)
		tmp = Float64(Float64(-100.0 / Float64(i / n)) * Float64(1.0 - t_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(100.0 * Float64(Float64(Float64(t_0 / Float64(1.0 / n)) - n) / i));
	else
		tmp = Float64(n * 100.0);
	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-61], N[(N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $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[(100.0 * N[(N[(N[(t$95$0 / N[(1.0 / n), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -1e-61

   1. Initial program 99.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 100.0%

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

   if -1e-61 < (/.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 21.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 91.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 43.3%

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

   4. Applied egg-rr 43.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{n}{i}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{i}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 43.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{i}\right)\right)\right)\right) \]

   6. Applied egg-rr 43.3%

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. remove-double-div N/A

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

      5. sub-div N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 92.2%

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

   8. Applied egg-rr 92.2%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \frac{\frac{t\_0}{\frac{1}{n}} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-125)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (/ (- (/ t_0 (/ 1.0 n)) n) i))
         (* n 100.0))))))

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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (((t_0 / (1.0 / n)) - n) / i);
	} else {
		tmp = n * 100.0;
	}
	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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((t_0 / (1.0 / n)) - n) / i);
	} else {
		tmp = n * 100.0;
	}
	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-125:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = 100.0 * (((t_0 / (1.0 / n)) - n) / i)
	else:
		tmp = n * 100.0
	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-125)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64(t_0 / Float64(1.0 / n)) - n) / i));
	else
		tmp = Float64(n * 100.0);
	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-125], N[(t$95$1 * 100.0), $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[(100.0 * N[(N[(N[(t$95$0 / N[(1.0 / n), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -2.00000000000000002e-125

   1. Initial program 97.4%

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

   if -2.00000000000000002e-125 < (/.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 20.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 75.4%

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

   5. Simplified 75.4%

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

   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 91.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 43.3%

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

   4. Applied egg-rr 43.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{n}{i}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{i}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 43.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{i}\right)\right)\right)\right) \]

   6. Applied egg-rr 43.3%

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. remove-double-div N/A

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

      5. sub-div N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 92.2%

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

   8. Applied egg-rr 92.2%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{t\_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-125)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (/ t_0 (/ i n)) (/ n i)))
         (* n 100.0))))))

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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = n * 100.0;
	}
	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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = n * 100.0;
	}
	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-125:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i))
	else:
		tmp = n * 100.0
	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-125)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 / Float64(i / n)) - Float64(n / i)));
	else
		tmp = Float64(n * 100.0);
	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-125], N[(t$95$1 * 100.0), $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[(100.0 * N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -2.00000000000000002e-125

   1. Initial program 97.4%

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

   if -2.00000000000000002e-125 < (/.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 20.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 75.4%

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

   5. Simplified 75.4%

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

   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 91.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 92.2%

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

   4. Applied egg-rr 92.2%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-125)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (* (/ -100.0 (/ i n)) (- 1.0 t_0))
         (* n 100.0))))))

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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-100.0 / (i / n)) * (1.0 - t_0);
	} else {
		tmp = n * 100.0;
	}
	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-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (-100.0 / (i / n)) * (1.0 - t_0);
	} else {
		tmp = n * 100.0;
	}
	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-125:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = (-100.0 / (i / n)) * (1.0 - t_0)
	else:
		tmp = n * 100.0
	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-125)
		tmp = Float64(t_1 * 100.0);
	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(i / n)) * Float64(1.0 - t_0));
	else
		tmp = Float64(n * 100.0);
	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-125], N[(t$95$1 * 100.0), $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[(i / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -2.00000000000000002e-125

   1. Initial program 97.4%

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

   if -2.00000000000000002e-125 < (/.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 20.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 75.4%

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

   5. Simplified 75.4%

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

   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 91.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 91.9%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{-100}{\frac{i}{n}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -2e-125)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY) (* n (* 100.0 (/ t_0 i))) (* n 100.0))))))

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 tmp;
	if (t_1 <= -2e-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (100.0 * (t_0 / i));
	} else {
		tmp = n * 100.0;
	}
	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 tmp;
	if (t_1 <= -2e-125) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (100.0 * (t_0 / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -2e-125:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_1 <= math.inf:
		tmp = n * (100.0 * (t_0 / i))
	else:
		tmp = n * 100.0
	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))
	tmp = 0.0
	if (t_1 <= -2e-125)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(100.0 * Float64(t_0 / i)));
	else
		tmp = Float64(n * 100.0);
	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]}, If[LessEqual[t$95$1, -2e-125], N[(t$95$1 * 100.0), $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 * N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -2.00000000000000002e-125

   1. Initial program 97.4%

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

   if -2.00000000000000002e-125 < (/.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 20.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 75.4%

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

   5. Simplified 75.4%

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

   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 91.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval 91.9%

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

   4. Applied egg-rr 91.9%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t\_0 \cdot 100\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -2e-125)
     t_1
     (if (<= t_0 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_0 INFINITY) t_1 (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-125) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-125) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -2e-125:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -2e-125)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-125], t$95$1, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n * 100.0), $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)) < -2.00000000000000002e-125 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 94.2%

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

   if -2.00000000000000002e-125 < (/.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 20.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 75.4%

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

   5. Simplified 75.4%

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

   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 i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{if}\;n \leq -7.6 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-194}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-217}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 2.15:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* 100.0 (* n (expm1 i))) i)))
   (if (<= n -7.6e+28)
     t_0
     (if (<= n -3.5e-194)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= n 9e-217)
         (* n (/ (+ -100.0 100.0) i))
         (if (<= n 2.15) (* 100.0 (/ i (/ i n))) t_0))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (n * expm1(i))) / i;
	double tmp;
	if (n <= -7.6e+28) {
		tmp = t_0;
	} else if (n <= -3.5e-194) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 9e-217) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 2.15) {
		tmp = 100.0 * (i / (i / n));
	} 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 tmp;
	if (n <= -7.6e+28) {
		tmp = t_0;
	} else if (n <= -3.5e-194) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 9e-217) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 2.15) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * (n * math.expm1(i))) / i
	tmp = 0
	if n <= -7.6e+28:
		tmp = t_0
	elif n <= -3.5e-194:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 9e-217:
		tmp = n * ((-100.0 + 100.0) / i)
	elif n <= 2.15:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(n * expm1(i))) / i)
	tmp = 0.0
	if (n <= -7.6e+28)
		tmp = t_0;
	elseif (n <= -3.5e-194)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 9e-217)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	elseif (n <= 2.15)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -7.6e+28], t$95$0, If[LessEqual[n, -3.5e-194], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9e-217], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.15], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -7.5999999999999998e28 or 2.14999999999999991 < n

   1. Initial program 25.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.6%

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

   5. Simplified 85.6%

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

   if -7.5999999999999998e28 < n < -3.5000000000000003e-194

   1. Initial program 21.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   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 73.0%

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

   5. Simplified 73.0%

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

   if -3.5000000000000003e-194 < n < 8.9999999999999997e-217

   1. Initial program 63.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.5%

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

   4. Applied egg-rr 84.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 8.9999999999999997e-217 < n < 2.14999999999999991

   1. Initial program 13.2%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-194}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-217}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 2.15:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{if}\;n \leq -156000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.02 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-214}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 2.7:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n (expm1 i)) (/ 100.0 i))))
   (if (<= n -156000.0)
     t_0
     (if (<= n -1.02e-199)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= n 3e-214)
         (* n (/ (+ -100.0 100.0) i))
         (if (<= n 2.7) (* 100.0 (/ i (/ i n))) t_0))))))

C

double code(double i, double n) {
	double t_0 = (n * expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -156000.0) {
		tmp = t_0;
	} else if (n <= -1.02e-199) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 3e-214) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 2.7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * Math.expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -156000.0) {
		tmp = t_0;
	} else if (n <= -1.02e-199) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 3e-214) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 2.7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * math.expm1(i)) * (100.0 / i)
	tmp = 0
	if n <= -156000.0:
		tmp = t_0
	elif n <= -1.02e-199:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 3e-214:
		tmp = n * ((-100.0 + 100.0) / i)
	elif n <= 2.7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * expm1(i)) * Float64(100.0 / i))
	tmp = 0.0
	if (n <= -156000.0)
		tmp = t_0;
	elseif (n <= -1.02e-199)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 3e-214)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	elseif (n <= 2.7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -156000.0], t$95$0, If[LessEqual[n, -1.02e-199], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-214], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -156000 or 2.7000000000000002 < n

   1. Initial program 26.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 84.8%

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

   5. Simplified 84.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. expm1-define N/A

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

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

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

      7. /-lowering-/.f64 84.6%

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

   7. Applied egg-rr 84.6%

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

   if -156000 < n < -1.02e-199

   1. Initial program 15.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 74.1%

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

   5. Simplified 74.1%

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

   if -1.02e-199 < n < 2.99999999999999994e-214

   1. Initial program 63.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.5%

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

   4. Applied egg-rr 84.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 2.99999999999999994e-214 < n < 2.7000000000000002

   1. Initial program 13.2%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -156000:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -1.02 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-214}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 2.7:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\\ \mathbf{if}\;n \leq -8 \cdot 10^{+95}:\\ \;\;\;\;n \cdot \left(100 + i \cdot t\_0\right)\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-178}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + t\_0 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (+
          50.0
          (*
           (* i -100.0)
           (+ (* i -0.041666666666666664) -0.16666666666666666)))))
   (if (<= n -8e+95)
     (* n (+ 100.0 (* i t_0)))
     (if (<= n -4.2e-195)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= n 1.06e-178)
         (* n (/ (+ -100.0 100.0) i))
         (+ (* n 100.0) (* t_0 (* i n))))))))

C

double code(double i, double n) {
	double t_0 = 50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666));
	double tmp;
	if (n <= -8e+95) {
		tmp = n * (100.0 + (i * t_0));
	} else if (n <= -4.2e-195) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 1.06e-178) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + (t_0 * (i * n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666));
	double tmp;
	if (n <= -8e+95) {
		tmp = n * (100.0 + (i * t_0));
	} else if (n <= -4.2e-195) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 1.06e-178) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + (t_0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))
	tmp = 0
	if n <= -8e+95:
		tmp = n * (100.0 + (i * t_0))
	elif n <= -4.2e-195:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 1.06e-178:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = (n * 100.0) + (t_0 * (i * n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666)))
	tmp = 0.0
	if (n <= -8e+95)
		tmp = Float64(n * Float64(100.0 + Float64(i * t_0)));
	elseif (n <= -4.2e-195)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 1.06e-178)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(t_0 * Float64(i * n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(50.0 + N[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8e+95], N[(n * N[(100.0 + N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.2e-195], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.06e-178], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(t$95$0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.00000000000000016e95

   1. Initial program 26.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 26.3%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 63.2%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 63.2%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 63.2%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)} \]

   if -8.00000000000000016e95 < n < -4.2e-195

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 73.1%

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

   5. Simplified 73.1%

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

   if -4.2e-195 < n < 1.05999999999999999e-178

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 1.05999999999999999e-178 < n

   1. Initial program 18.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(n \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\left(i \cdot n\right) \cdot \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(i \cdot n\right), \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(n \cdot i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   10. Simplified 75.9%

       \[\leadsto 100 \cdot n + \color{blue}{\left(n \cdot i\right) \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+95}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-178}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(\left(\frac{\left(i \cdot -100\right) \cdot \left(0.5 + i \cdot 0.25\right) + \left(\frac{100 \cdot \left(i \cdot \left(0.3333333333333333 + i \cdot 0.4583333333333333\right)\right)}{n} + -50\right)}{n} - \left(i \cdot -100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right) - -50\right)\right)\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.2e-192)
   (+
    (* n 100.0)
    (*
     i
     (*
      n
      (-
       (-
        (/
         (+
          (* (* i -100.0) (+ 0.5 (* i 0.25)))
          (+
           (/
            (* 100.0 (* i (+ 0.3333333333333333 (* i 0.4583333333333333))))
            n)
           -50.0))
         n)
        (* (* i -100.0) (+ 0.16666666666666666 (* i 0.041666666666666664))))
       -50.0))))
   (if (<= n 8.2e-179)
     (* n (/ (+ -100.0 100.0) i))
     (+
      (* n 100.0)
      (*
       (+
        50.0
        (* (* i -100.0) (+ (* i -0.041666666666666664) -0.16666666666666666)))
       (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.2e-192) {
		tmp = (n * 100.0) + (i * (n * ((((((i * -100.0) * (0.5 + (i * 0.25))) + (((100.0 * (i * (0.3333333333333333 + (i * 0.4583333333333333)))) / n) + -50.0)) / n) - ((i * -100.0) * (0.16666666666666666 + (i * 0.041666666666666664)))) - -50.0)));
	} else if (n <= 8.2e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.2d-192)) then
        tmp = (n * 100.0d0) + (i * (n * ((((((i * (-100.0d0)) * (0.5d0 + (i * 0.25d0))) + (((100.0d0 * (i * (0.3333333333333333d0 + (i * 0.4583333333333333d0)))) / n) + (-50.0d0))) / n) - ((i * (-100.0d0)) * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))) - (-50.0d0))))
    else if (n <= 8.2d-179) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else
        tmp = (n * 100.0d0) + ((50.0d0 + ((i * (-100.0d0)) * ((i * (-0.041666666666666664d0)) + (-0.16666666666666666d0)))) * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.2e-192) {
		tmp = (n * 100.0) + (i * (n * ((((((i * -100.0) * (0.5 + (i * 0.25))) + (((100.0 * (i * (0.3333333333333333 + (i * 0.4583333333333333)))) / n) + -50.0)) / n) - ((i * -100.0) * (0.16666666666666666 + (i * 0.041666666666666664)))) - -50.0)));
	} else if (n <= 8.2e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.2e-192:
		tmp = (n * 100.0) + (i * (n * ((((((i * -100.0) * (0.5 + (i * 0.25))) + (((100.0 * (i * (0.3333333333333333 + (i * 0.4583333333333333)))) / n) + -50.0)) / n) - ((i * -100.0) * (0.16666666666666666 + (i * 0.041666666666666664)))) - -50.0)))
	elif n <= 8.2e-179:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.2e-192)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(Float64(Float64(Float64(Float64(Float64(i * -100.0) * Float64(0.5 + Float64(i * 0.25))) + Float64(Float64(Float64(100.0 * Float64(i * Float64(0.3333333333333333 + Float64(i * 0.4583333333333333)))) / n) + -50.0)) / n) - Float64(Float64(i * -100.0) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))) - -50.0))));
	elseif (n <= 8.2e-179)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666))) * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.2e-192)
		tmp = (n * 100.0) + (i * (n * ((((((i * -100.0) * (0.5 + (i * 0.25))) + (((100.0 * (i * (0.3333333333333333 + (i * 0.4583333333333333)))) / n) + -50.0)) / n) - ((i * -100.0) * (0.16666666666666666 + (i * 0.041666666666666664)))) - -50.0)));
	elseif (n <= 8.2e-179)
		tmp = n * ((-100.0 + 100.0) / i);
	else
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.2e-192], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(N[(N[(N[(N[(N[(i * -100.0), $MachinePrecision] * N[(0.5 + N[(i * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(100.0 * N[(i * N[(0.3333333333333333 + N[(i * 0.4583333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + -50.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(N[(i * -100.0), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.2e-179], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(50.0 + N[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.2000000000000003e-192

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 26.4%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 53.4%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(-1 \cdot \left(n \cdot \left(\left(-100 \cdot \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + -1 \cdot \frac{\left(-100 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot i\right)\right) + 100 \cdot \frac{i \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)}{n}\right) - 50}{n}\right) - 50\right)\right)\right)}\right)\right) \]

   10. Simplified 62.1%

       \[\leadsto 100 \cdot n + i \cdot \color{blue}{\left(\left(\left(\left(-100 \cdot i\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right) - \frac{\left(-100 \cdot i\right) \cdot \left(0.5 + i \cdot 0.25\right) + \left(\frac{100 \cdot \left(i \cdot \left(0.3333333333333333 + i \cdot 0.4583333333333333\right)\right)}{n} + -50\right)}{n}\right) + -50\right) \cdot \left(0 - n\right)\right)} \]

   if -5.2000000000000003e-192 < n < 8.2e-179

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 8.2e-179 < n

   1. Initial program 18.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(n \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\left(i \cdot n\right) \cdot \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(i \cdot n\right), \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(n \cdot i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   10. Simplified 75.9%

       \[\leadsto 100 \cdot n + \color{blue}{\left(n \cdot i\right) \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(\left(\frac{\left(i \cdot -100\right) \cdot \left(0.5 + i \cdot 0.25\right) + \left(\frac{100 \cdot \left(i \cdot \left(0.3333333333333333 + i \cdot 0.4583333333333333\right)\right)}{n} + -50\right)}{n} - \left(i \cdot -100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right) - -50\right)\right)\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-199}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot 100\right) \cdot \left(n \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-178}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.65e-199)
   (+
    (* n 100.0)
    (*
     i
     (+
      (* (+ (/ 0.5 n) -0.5) (* n -100.0))
      (*
       (* i 100.0)
       (* n (+ 0.16666666666666666 (* i 0.041666666666666664)))))))
   (if (<= n 1.15e-178)
     (* n (/ (+ -100.0 100.0) i))
     (+
      (* n 100.0)
      (*
       (+
        50.0
        (* (* i -100.0) (+ (* i -0.041666666666666664) -0.16666666666666666)))
       (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.65e-199) {
		tmp = (n * 100.0) + (i * ((((0.5 / n) + -0.5) * (n * -100.0)) + ((i * 100.0) * (n * (0.16666666666666666 + (i * 0.041666666666666664))))));
	} else if (n <= 1.15e-178) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.65d-199)) then
        tmp = (n * 100.0d0) + (i * ((((0.5d0 / n) + (-0.5d0)) * (n * (-100.0d0))) + ((i * 100.0d0) * (n * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))
    else if (n <= 1.15d-178) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else
        tmp = (n * 100.0d0) + ((50.0d0 + ((i * (-100.0d0)) * ((i * (-0.041666666666666664d0)) + (-0.16666666666666666d0)))) * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.65e-199) {
		tmp = (n * 100.0) + (i * ((((0.5 / n) + -0.5) * (n * -100.0)) + ((i * 100.0) * (n * (0.16666666666666666 + (i * 0.041666666666666664))))));
	} else if (n <= 1.15e-178) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.65e-199:
		tmp = (n * 100.0) + (i * ((((0.5 / n) + -0.5) * (n * -100.0)) + ((i * 100.0) * (n * (0.16666666666666666 + (i * 0.041666666666666664))))))
	elif n <= 1.15e-178:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.65e-199)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(Float64(Float64(0.5 / n) + -0.5) * Float64(n * -100.0)) + Float64(Float64(i * 100.0) * Float64(n * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))));
	elseif (n <= 1.15e-178)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666))) * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.65e-199)
		tmp = (n * 100.0) + (i * ((((0.5 / n) + -0.5) * (n * -100.0)) + ((i * 100.0) * (n * (0.16666666666666666 + (i * 0.041666666666666664))))));
	elseif (n <= 1.15e-178)
		tmp = n * ((-100.0 + 100.0) / i);
	else
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.65e-199], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] * N[(n * -100.0), $MachinePrecision]), $MachinePrecision] + N[(N[(i * 100.0), $MachinePrecision] * N[(n * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-178], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(50.0 + N[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6500000000000001e-199

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 26.4%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 53.4%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \color{blue}{\left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \left(\left(100 \cdot i\right) \cdot \color{blue}{\left(n \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\left(100 \cdot i\right), \color{blue}{\left(n \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \left(\color{blue}{n} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(n, -100\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 61.7%

       \[\leadsto 100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \color{blue}{\left(100 \cdot i\right) \cdot \left(n \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)}\right) \]

   if -1.6500000000000001e-199 < n < 1.14999999999999997e-178

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 1.14999999999999997e-178 < n

   1. Initial program 18.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(n \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\left(i \cdot n\right) \cdot \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(i \cdot n\right), \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(n \cdot i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   10. Simplified 75.9%

       \[\leadsto 100 \cdot n + \color{blue}{\left(n \cdot i\right) \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-199}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot 100\right) \cdot \left(n \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-178}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-193}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 9.4 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.7e-193)
   (+
    (* n 100.0)
    (*
     i
     (+
      (* n 50.0)
      (* i (* n (+ (* i 4.166666666666667) 16.666666666666668))))))
   (if (<= n 9.4e-180)
     (* n (/ (+ -100.0 100.0) i))
     (+
      (* n 100.0)
      (*
       (+
        50.0
        (* (* i -100.0) (+ (* i -0.041666666666666664) -0.16666666666666666)))
       (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.7e-193) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else if (n <= 9.4e-180) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.7d-193)) then
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) + (i * (n * ((i * 4.166666666666667d0) + 16.666666666666668d0)))))
    else if (n <= 9.4d-180) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else
        tmp = (n * 100.0d0) + ((50.0d0 + ((i * (-100.0d0)) * ((i * (-0.041666666666666664d0)) + (-0.16666666666666666d0)))) * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.7e-193) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else if (n <= 9.4e-180) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.7e-193:
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))))
	elif n <= 9.4e-180:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.7e-193)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(i * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))))));
	elseif (n <= 9.4e-180)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666))) * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.7e-193)
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	elseif (n <= 9.4e-180)
		tmp = n * ((-100.0 + 100.0) / i);
	else
		tmp = (n * 100.0) + ((50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666))) * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.7e-193], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(i * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.4e-180], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(50.0 + N[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6999999999999999e-193

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(50 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.7%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(4.166666666666667 \cdot i + 16.666666666666668\right)\right)\right)} \]

   if -2.6999999999999999e-193 < n < 9.39999999999999951e-180

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 9.39999999999999951e-180 < n

   1. Initial program 18.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(n \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\left(i \cdot n\right) \cdot \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(i \cdot n\right), \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\left(n \cdot i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \left(\color{blue}{50} + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, i\right), \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   10. Simplified 75.9%

       \[\leadsto 100 \cdot n + \color{blue}{\left(n \cdot i\right) \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-193}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 9.4 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{-196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.12 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.1e-196)
   (+
    (* n 100.0)
    (*
     i
     (+
      (* n 50.0)
      (* i (* n (+ (* i 4.166666666666667) 16.666666666666668))))))
   (if (<= n 1.12e-179)
     (* n (/ (+ -100.0 100.0) i))
     (*
      n
      (+
       100.0
       (*
        i
        (+
         50.0
         (*
          (* i -100.0)
          (+ (* i -0.041666666666666664) -0.16666666666666666)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.1e-196) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else if (n <= 1.12e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.1d-196)) then
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) + (i * (n * ((i * 4.166666666666667d0) + 16.666666666666668d0)))))
    else if (n <= 1.12d-179) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + ((i * (-100.0d0)) * ((i * (-0.041666666666666664d0)) + (-0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.1e-196) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else if (n <= 1.12e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.1e-196:
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))))
	elif n <= 1.12e-179:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.1e-196)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(i * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))))));
	elseif (n <= 1.12e-179)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.1e-196)
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	elseif (n <= 1.12e-179)
		tmp = n * ((-100.0 + 100.0) / i);
	else
		tmp = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.1e-196], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(i * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.12e-179], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.09999999999999988e-196

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(50 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 61.7%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(4.166666666666667 \cdot i + 16.666666666666668\right)\right)\right)} \]

   if -2.09999999999999988e-196 < n < 1.11999999999999999e-179

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 1.11999999999999999e-179 < n

   1. Initial program 18.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 75.9%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{-196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.12 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (+
           100.0
           (*
            i
            (+
             50.0
             (*
              (* i -100.0)
              (+ (* i -0.041666666666666664) -0.16666666666666666))))))))
   (if (<= n -5.2e-197)
     t_0
     (if (<= n 2.75e-179) (* n (/ (+ -100.0 100.0) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	double tmp;
	if (n <= -5.2e-197) {
		tmp = t_0;
	} else if (n <= 2.75e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} 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 * (-100.0d0)) * ((i * (-0.041666666666666664d0)) + (-0.16666666666666666d0))))))
    if (n <= (-5.2d-197)) then
        tmp = t_0
    else if (n <= 2.75d-179) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    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 * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	double tmp;
	if (n <= -5.2e-197) {
		tmp = t_0;
	} else if (n <= 2.75e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))))
	tmp = 0
	if n <= -5.2e-197:
		tmp = t_0
	elif n <= 2.75e-179:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(Float64(i * -100.0) * Float64(Float64(i * -0.041666666666666664) + -0.16666666666666666))))))
	tmp = 0.0
	if (n <= -5.2e-197)
		tmp = t_0;
	elseif (n <= 2.75e-179)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + ((i * -100.0) * ((i * -0.041666666666666664) + -0.16666666666666666)))));
	tmp = 0.0;
	if (n <= -5.2e-197)
		tmp = t_0;
	elseif (n <= 2.75e-179)
		tmp = n * ((-100.0 + 100.0) / i);
	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[(N[(i * -100.0), $MachinePrecision] * N[(N[(i * -0.041666666666666664), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e-197], t$95$0, If[LessEqual[n, 2.75e-179], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.2000000000000003e-197 or 2.7500000000000001e-179 < n

   1. Initial program 22.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 22.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(-100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)\right) + i \cdot \left(-100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right) - \left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right) + -100 \cdot \left(n \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right)\right)\right)} \]

   7. Simplified 60.8%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\left(\frac{0.5}{n} + -0.5\right) \cdot \left(n \cdot -100\right) + \left(i \cdot -100\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{0.25}{n} + \left(\frac{0.25}{n \cdot \left(n \cdot n\right)} - \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right)\right)\right) + n \cdot \left(\left(\frac{0.5}{n} + -0.16666666666666666\right) - \frac{0.3333333333333333}{n \cdot n}\right)\right)\right)} \]

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + -100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(-100 \cdot \left(i \cdot \left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(\left(-100 \cdot i\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\left(-100 \cdot i\right), \color{blue}{\left(\frac{-1}{24} \cdot i - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\color{blue}{\frac{-1}{24} \cdot i} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \left(\frac{-1}{24} \cdot i + \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(\frac{-1}{24} \cdot i\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, i\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{24}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \left(-100 \cdot i\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)} \]

   if -5.2000000000000003e-197 < n < 2.7500000000000001e-179

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-197}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot -100\right) \cdot \left(i \cdot -0.041666666666666664 + -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-198}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.3e-198)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 1.4e-219)
     (* n (/ (+ -100.0 100.0) i))
     (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 <= -4.3e-198) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.4e-219) {
		tmp = n * ((-100.0 + 100.0) / i);
	} 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 <= (-4.3d-198)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 1.4d-219) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    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 <= -4.3e-198) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.4e-219) {
		tmp = n * ((-100.0 + 100.0) / i);
	} 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 <= -4.3e-198:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 1.4e-219:
		tmp = n * ((-100.0 + 100.0) / i)
	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 <= -4.3e-198)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 1.4e-219)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	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 <= -4.3e-198)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 1.4e-219)
		tmp = n * ((-100.0 + 100.0) / i);
	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, -4.3e-198], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-219], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $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 < -4.3000000000000003e-198

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. *-lowering-*.f64 61.3%

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

   8. Simplified 61.3%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(n \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]

   if -4.3000000000000003e-198 < n < 1.39999999999999995e-219

   1. Initial program 63.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.5%

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

   4. Applied egg-rr 84.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 1.39999999999999995e-219 < n < 1.55000000000000004

   1. Initial program 13.2%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.55000000000000004 < n

   1. Initial program 21.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.0%

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

   5. Simplified 88.0%

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

   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right)\right), i\right) \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + 50 \cdot i\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-198}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-194}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.2e-194)
   (* 100.0 (* n (+ 1.0 (* 0.16666666666666666 (* i i)))))
   (if (<= n 5.6e-219)
     (* n (/ (+ -100.0 100.0) i))
     (if (<= n 1.5)
       (* 100.0 (/ i (/ i n)))
       (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.2e-194) {
		tmp = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	} else if (n <= 5.6e-219) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 1.5) {
		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 <= (-7.2d-194)) then
        tmp = 100.0d0 * (n * (1.0d0 + (0.16666666666666666d0 * (i * i))))
    else if (n <= 5.6d-219) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else if (n <= 1.5d0) 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 <= -7.2e-194) {
		tmp = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	} else if (n <= 5.6e-219) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else if (n <= 1.5) {
		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 <= -7.2e-194:
		tmp = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))))
	elif n <= 5.6e-219:
		tmp = n * ((-100.0 + 100.0) / i)
	elif n <= 1.5:
		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 <= -7.2e-194)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(0.16666666666666666 * Float64(i * i)))));
	elseif (n <= 5.6e-219)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	elseif (n <= 1.5)
		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 <= -7.2e-194)
		tmp = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	elseif (n <= 5.6e-219)
		tmp = n * ((-100.0 + 100.0) / i);
	elseif (n <= 1.5)
		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, -7.2e-194], N[(100.0 * N[(n * N[(1.0 + N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-219], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], 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 < -7.2e-194

   1. Initial program 26.4%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) + \color{blue}{i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \left(\color{blue}{i} \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 57.9%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right) + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \mathsf{*.f64}\left(n, i\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 57.9%

      \[\leadsto 100 \cdot \left(n + i \cdot \left(\color{blue}{-0.5} + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right) \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \frac{1}{6} \cdot {i}^{2}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n \cdot \left(1 + \frac{1}{6} \cdot {i}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + \frac{1}{6} \cdot {i}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {i}^{2}\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \left({i}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(i \cdot i\right), \frac{1}{6}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{6}\right)\right)\right)\right) \]

   11. Simplified 61.3%

       \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \left(i \cdot i\right) \cdot 0.16666666666666666\right)\right)} \]

   if -7.2e-194 < n < 5.5999999999999998e-219

   1. Initial program 63.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.5%

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

   4. Applied egg-rr 84.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]

   if 5.5999999999999998e-219 < n < 1.5

   1. Initial program 13.2%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.5 < n

   1. Initial program 21.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.0%

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

   5. Simplified 88.0%

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

   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right)\right), i\right) \]

   8. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + 50 \cdot i\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-194}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \left(1 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{if}\;n \leq -1.26 \cdot 10^{-191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (+ 1.0 (* 0.16666666666666666 (* i i)))))))
   (if (<= n -1.26e-191)
     t_0
     (if (<= n 2.9e-179) (* n (/ (+ -100.0 100.0) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	double tmp;
	if (n <= -1.26e-191) {
		tmp = t_0;
	} else if (n <= 2.9e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} 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 * (n * (1.0d0 + (0.16666666666666666d0 * (i * i))))
    if (n <= (-1.26d-191)) then
        tmp = t_0
    else if (n <= 2.9d-179) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	double tmp;
	if (n <= -1.26e-191) {
		tmp = t_0;
	} else if (n <= 2.9e-179) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))))
	tmp = 0
	if n <= -1.26e-191:
		tmp = t_0
	elif n <= 2.9e-179:
		tmp = n * ((-100.0 + 100.0) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(1.0 + Float64(0.16666666666666666 * Float64(i * i)))))
	tmp = 0.0
	if (n <= -1.26e-191)
		tmp = t_0;
	elseif (n <= 2.9e-179)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n * (1.0 + (0.16666666666666666 * (i * i))));
	tmp = 0.0;
	if (n <= -1.26e-191)
		tmp = t_0;
	elseif (n <= 2.9e-179)
		tmp = n * ((-100.0 + 100.0) / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(1.0 + N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.26e-191], t$95$0, If[LessEqual[n, 2.9e-179], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.26e-191 or 2.8999999999999999e-179 < n

   1. Initial program 22.5%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) + \color{blue}{i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \left(\color{blue}{i} \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 61.4%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right) + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \mathsf{*.f64}\left(n, i\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 61.4%

      \[\leadsto 100 \cdot \left(n + i \cdot \left(\color{blue}{-0.5} + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right) \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \frac{1}{6} \cdot {i}^{2}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n \cdot \left(1 + \frac{1}{6} \cdot {i}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + \frac{1}{6} \cdot {i}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {i}^{2}\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \left({i}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(i \cdot i\right), \frac{1}{6}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 64.9%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \frac{1}{6}\right)\right)\right)\right) \]

   11. Simplified 64.9%

       \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \left(i \cdot i\right) \cdot 0.16666666666666666\right)\right)} \]

   if -1.26e-191 < n < 2.8999999999999999e-179

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-191}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -1.2e-199)
     t_0
     (if (<= n 8e-180) (* n (/ (+ -100.0 100.0) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.2e-199) {
		tmp = t_0;
	} else if (n <= 8e-180) {
		tmp = n * ((-100.0 + 100.0) / i);
	} 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 <= (-1.2d-199)) then
        tmp = t_0
    else if (n <= 8d-180) then
        tmp = n * (((-100.0d0) + 100.0d0) / i)
    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 <= -1.2e-199) {
		tmp = t_0;
	} else if (n <= 8e-180) {
		tmp = n * ((-100.0 + 100.0) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -1.2e-199:
		tmp = t_0
	elif n <= 8e-180:
		tmp = n * ((-100.0 + 100.0) / i)
	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 <= -1.2e-199)
		tmp = t_0;
	elseif (n <= 8e-180)
		tmp = Float64(n * Float64(Float64(-100.0 + 100.0) / i));
	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 <= -1.2e-199)
		tmp = t_0;
	elseif (n <= 8e-180)
		tmp = n * ((-100.0 + 100.0) / i);
	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, -1.2e-199], t$95$0, If[LessEqual[n, 8e-180], N[(n * N[(N[(-100.0 + 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.19999999999999998e-199 or 8.0000000000000002e-180 < n

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 71.0%

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

   5. Simplified 71.0%

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

   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 62.2%

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

   8. Simplified 62.2%

      \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   if -1.19999999999999998e-199 < n < 8.0000000000000002e-180

   1. Initial program 56.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 80.3%

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

   4. Applied egg-rr 80.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in i around 0

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

   9. Simplified 75.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{i} \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-199}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \frac{-100 + 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -2.7e+66) t_0 (if (<= n 1.42) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -2.7e+66) {
		tmp = t_0;
	} else if (n <= 1.42) {
		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 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-2.7d+66)) then
        tmp = t_0
    else if (n <= 1.42d0) 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 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -2.7e+66) {
		tmp = t_0;
	} else if (n <= 1.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -2.7e+66:
		tmp = t_0
	elif n <= 1.42:
		tmp = 100.0 * (i / (i / n))
	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 <= -2.7e+66)
		tmp = t_0;
	elseif (n <= 1.42)
		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 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -2.7e+66)
		tmp = t_0;
	elseif (n <= 1.42)
		tmp = 100.0 * (i / (i / n));
	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, -2.7e+66], t$95$0, If[LessEqual[n, 1.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.7e66 or 1.4199999999999999 < n

   1. Initial program 24.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.6%

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

   5. Simplified 85.6%

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

   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 62.0%

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

   8. Simplified 62.0%

      \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   if -2.7e66 < n < 1.4199999999999999

   1. Initial program 32.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 60.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+66}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{-20}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= i -1.75e-26)
     t_0
     (if (<= i 1.22e-20) (* 100.0 (+ n (* i -0.5))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (i <= -1.75e-26) {
		tmp = t_0;
	} else if (i <= 1.22e-20) {
		tmp = 100.0 * (n + (i * -0.5));
	} 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 / (i / n))
    if (i <= (-1.75d-26)) then
        tmp = t_0
    else if (i <= 1.22d-20) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    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 / (i / n));
	double tmp;
	if (i <= -1.75e-26) {
		tmp = t_0;
	} else if (i <= 1.22e-20) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if i <= -1.75e-26:
		tmp = t_0
	elif i <= 1.22e-20:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (i <= -1.75e-26)
		tmp = t_0;
	elseif (i <= 1.22e-20)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (i <= -1.75e-26)
		tmp = t_0;
	elseif (i <= 1.22e-20)
		tmp = 100.0 * (n + (i * -0.5));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.75e-26], t$95$0, If[LessEqual[i, 1.22e-20], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.74999999999999992e-26 or 1.22000000000000003e-20 < i

   1. Initial program 50.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 20.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -1.74999999999999992e-26 < i < 1.22000000000000003e-20

   1. Initial program 9.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) + \color{blue}{i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \left(\color{blue}{i} \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 75.5%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right) + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \mathsf{*.f64}\left(n, i\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 75.5%

      \[\leadsto 100 \cdot \left(n + i \cdot \left(\color{blue}{-0.5} + \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) \cdot \left(n \cdot i\right)\right)\right) \]

   9. Taylor expanded in i around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 87.0%

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

   11. Simplified 87.0%

       \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 54.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.95e-16) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.95e-16) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.95d-16) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.95e-16) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.95e-16:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.95e-16)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.95e-16)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.95e-16], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.94999999999999989e-16

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 65.9%

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

   5. Simplified 65.9%

      \[\leadsto \color{blue}{100 \cdot n} \]

   if 1.94999999999999989e-16 < i

   1. Initial program 47.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 42.5%

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

   5. Simplified 42.5%

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

   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. *-lowering-*.f64 33.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right)\right), i\right) \]

   8. Simplified 33.6%

      \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + 50 \cdot i\right)\right)}}{i} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   10. Step-by-step derivation

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 21.4%

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

   11. Simplified 21.4%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 50.0% 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 28.6%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 49.8%

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

5. Simplified 49.8%

   \[\leadsto \color{blue}{100 \cdot n} \]

6. Final simplification 49.8%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer Target 1: 34.2% 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 2024156 
(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))))