Compound Interest

Percentage Accurate: 29.0% → 99.5%

Time: 17.5s

Alternatives: 17

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: 29.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{t\_0}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \end{array} \end{array} \]

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 -4e-29)
     (* 100.0 (/ (+ (* (/ i n) (/ t_0 i)) (/ -1.0 n)) (/ (/ i n) n)))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (* t_1 100.0)
         (/ 100.0 (+ (/ 1.0 n) (* i (+ (/ 0.5 (* n n)) (/ -0.5 n))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -4e-29) {
		tmp = 100.0 * ((((i / n) * (t_0 / i)) + (-1.0 / n)) / ((i / n) / n));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -4e-29) {
		tmp = 100.0 * ((((i / n) * (t_0 / i)) + (-1.0 / n)) / ((i / n) / n));
	} 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 = t_1 * 100.0;
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -4e-29:
		tmp = 100.0 * ((((i / n) * (t_0 / i)) + (-1.0 / n)) / ((i / n) / n))
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = t_1 * 100.0
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -4e-29)
		tmp = Float64(100.0 * Float64(Float64(Float64(Float64(i / n) * Float64(t_0 / i)) + Float64(-1.0 / n)) / Float64(Float64(i / n) / n)));
	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(t_1 * 100.0);
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-29], N[(100.0 * N[(N[(N[(N[(i / n), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / N[(N[(i / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$1 * 100.0), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -3.99999999999999977e-29

   1. Initial program 99.9%

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

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

      3. associate-/r* N/A

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

      4. frac-sub N/A

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

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

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

   4. Applied egg-rr 100.0%

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

   if -3.99999999999999977e-29 < (/.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 24.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 99.6%

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

   4. Applied egg-rr 99.6%

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

   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 100.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

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

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

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

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

      11. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -0.00072:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -0.00072)
     t_0
     (if (<= n 3.8e-26)
       (/ 100.0 (+ (/ 1.0 n) (* i (+ (/ 0.5 (* n n)) (/ -0.5 n)))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -0.00072) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -0.00072) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -0.00072:
		tmp = t_0
	elif n <= 3.8e-26:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -0.00072)
		tmp = t_0;
	elseif (n <= 3.8e-26)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.00072], t$95$0, If[LessEqual[n, 3.8e-26], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.20000000000000045e-4 or 3.80000000000000015e-26 < n

   1. Initial program 26.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 92.4%

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

   7. Simplified 92.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. expm1-define N/A

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

      9. expm1-lowering-expm1.f64 92.5%

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

   9. Applied egg-rr 92.5%

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

   if -7.20000000000000045e-4 < n < 3.80000000000000015e-26

   1. Initial program 33.7%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

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

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

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

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

      11. metadata-eval 33.7%

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

   4. Applied egg-rr 33.7%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 68.2%

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

   7. Simplified 68.2%

      \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.00072:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -0.00105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -0.00105)
     t_0
     (if (<= n 3.8e-26)
       (/ 100.0 (+ (/ 1.0 n) (* i (+ (/ 0.5 (* n n)) (/ -0.5 n)))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -0.00105) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / 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 <= -0.00105) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / 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 <= -0.00105:
		tmp = t_0
	elif n <= 3.8e-26:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -0.00105)
		tmp = t_0;
	elseif (n <= 3.8e-26)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.00105], t$95$0, If[LessEqual[n, 3.8e-26], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -0.00104999999999999994 or 3.80000000000000015e-26 < n

   1. Initial program 26.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. expm1-define N/A

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

      4. expm1-lowering-expm1.f64 92.4%

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

   5. Simplified 92.4%

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

   if -0.00104999999999999994 < n < 3.80000000000000015e-26

   1. Initial program 33.7%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

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

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

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

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

      11. metadata-eval 33.7%

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

   4. Applied egg-rr 33.7%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 68.2%

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

   7. Simplified 68.2%

      \[\leadsto \frac{100}{\color{blue}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e+60)
   (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))
   (if (<= n -1.75e-218)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.35e-222)
       (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n)))
       (if (<= n 3.8e-26)
         (* i (/ (* n 100.0) i))
         (*
          n
          (+
           100.0
           (*
            i
            (+
             50.0
             (*
              (* i 100.0)
              (+ 0.16666666666666666 (* i 0.041666666666666664))))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e+60) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= -1.75e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.35e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 3.8e-26) {
		tmp = i * ((n * 100.0) / i);
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9d+60)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    else if (n <= (-1.75d-218)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.35d-222) then
        tmp = 100.0d0 * ((1.0d0 / n) / ((1.0d0 / n) / n))
    else if (n <= 3.8d-26) then
        tmp = i * ((n * 100.0d0) / i)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + ((i * 100.0d0) * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9e+60) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= -1.75e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.35e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 3.8e-26) {
		tmp = i * ((n * 100.0) / i);
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9e+60:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	elif n <= -1.75e-218:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.35e-222:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	elif n <= 3.8e-26:
		tmp = i * ((n * 100.0) / i)
	else:
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e+60)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	elseif (n <= -1.75e-218)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.35e-222)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	elseif (n <= 3.8e-26)
		tmp = Float64(i * Float64(Float64(n * 100.0) / i));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(Float64(i * 100.0) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9e+60)
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	elseif (n <= -1.75e-218)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.35e-222)
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	elseif (n <= 3.8e-26)
		tmp = i * ((n * 100.0) / i);
	else
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e+60], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.75e-218], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.35e-222], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-26], N[(i * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -9.00000000000000026e60

   1. Initial program 21.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 60.4%

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

   10. Simplified 60.4%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)} \]

   if -9.00000000000000026e60 < n < -1.75e-218

   1. Initial program 31.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 64.4%

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

   if -1.75e-218 < n < 2.3499999999999999e-222

   1. Initial program 76.6%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. sub-div N/A

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

      5. frac-sub N/A

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

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

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

   4. Applied egg-rr 45.8%

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

   5. Taylor expanded in i around 0

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

      1. /-lowering-/.f64 80.7%

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

   7. Simplified 80.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{n}}}{\frac{\frac{1}{n}}{n}} \]

   if 2.3499999999999999e-222 < n < 3.80000000000000015e-26

   1. Initial program 8.0%

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

   3. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 3.0%

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

   5. Simplified 3.0%

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

   6. Taylor expanded in i around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. /-lowering-/.f64 49.5%

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

   8. Simplified 49.5%

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

   9. Taylor expanded in i around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-lowering-*.f64 68.5%

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

   11. Simplified 68.5%

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

   if 3.80000000000000015e-26 < n

   1. Initial program 29.5%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 77.0%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 77.0%

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

   8. Simplified 77.0%

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \left(100 \cdot i\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-40}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + 0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.8e+61)
   (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))
   (if (<= n -2.2e-218)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 5.3e-222)
       (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n)))
       (if (<= n 7e-40)
         (* i (/ (* n 100.0) i))
         (* (* n 100.0) (+ 1.0 (* 0.041666666666666664 (* i (* i i))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.8e+61) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= -2.2e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.3e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 7e-40) {
		tmp = i * ((n * 100.0) / i);
	} else {
		tmp = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * 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 <= (-4.8d+61)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    else if (n <= (-2.2d-218)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.3d-222) then
        tmp = 100.0d0 * ((1.0d0 / n) / ((1.0d0 / n) / n))
    else if (n <= 7d-40) then
        tmp = i * ((n * 100.0d0) / i)
    else
        tmp = (n * 100.0d0) * (1.0d0 + (0.041666666666666664d0 * (i * (i * i))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.8e+61) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= -2.2e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.3e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 7e-40) {
		tmp = i * ((n * 100.0) / i);
	} else {
		tmp = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.8e+61:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	elif n <= -2.2e-218:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.3e-222:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	elif n <= 7e-40:
		tmp = i * ((n * 100.0) / i)
	else:
		tmp = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.8e+61)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	elseif (n <= -2.2e-218)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.3e-222)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	elseif (n <= 7e-40)
		tmp = Float64(i * Float64(Float64(n * 100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(0.041666666666666664 * Float64(i * Float64(i * i)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.8e+61)
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	elseif (n <= -2.2e-218)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.3e-222)
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	elseif (n <= 7e-40)
		tmp = i * ((n * 100.0) / i);
	else
		tmp = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.8e+61], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.2e-218], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-222], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-40], N[(i * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(0.041666666666666664 * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.7999999999999998e61

   1. Initial program 21.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 60.4%

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

   10. Simplified 60.4%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)} \]

   if -4.7999999999999998e61 < n < -2.20000000000000007e-218

   1. Initial program 31.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 64.4%

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

   if -2.20000000000000007e-218 < n < 5.29999999999999981e-222

   1. Initial program 76.6%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. sub-div N/A

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

      5. frac-sub N/A

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

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

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

   4. Applied egg-rr 45.8%

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

   5. Taylor expanded in i around 0

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

      1. /-lowering-/.f64 80.7%

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

   7. Simplified 80.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{n}}}{\frac{\frac{1}{n}}{n}} \]

   if 5.29999999999999981e-222 < n < 7.0000000000000003e-40

   1. Initial program 8.3%

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

   3. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 3.0%

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

   5. Simplified 3.0%

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

   6. Taylor expanded in i around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. /-lowering-/.f64 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in i around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-lowering-*.f64 69.4%

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

   11. Simplified 69.4%

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

   if 7.0000000000000003e-40 < n

   1. Initial program 28.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 72.6%

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]

   11. Taylor expanded in i around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {i}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({i}^{3}\right)}\right)\right)\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 74.9%

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

   13. Simplified 74.9%

       \[\leadsto \left(100 \cdot n\right) \cdot \left(1 + \color{blue}{0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-40}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + 0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \left(1 + 0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{if}\;n \leq -9 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (+ 1.0 (* 0.041666666666666664 (* i (* i i)))))))
   (if (<= n -9e+60)
     t_0
     (if (<= n -6e-218)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 5.5e-222)
         (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n)))
         (if (<= n 1.85e-39) (* i (/ (* n 100.0) i)) t_0))))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))));
	double tmp;
	if (n <= -9e+60) {
		tmp = t_0;
	} else if (n <= -6e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.5e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 1.85e-39) {
		tmp = i * ((n * 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) * (1.0d0 + (0.041666666666666664d0 * (i * (i * i))))
    if (n <= (-9d+60)) then
        tmp = t_0
    else if (n <= (-6d-218)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.5d-222) then
        tmp = 100.0d0 * ((1.0d0 / n) / ((1.0d0 / n) / n))
    else if (n <= 1.85d-39) then
        tmp = i * ((n * 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) * (1.0 + (0.041666666666666664 * (i * (i * i))));
	double tmp;
	if (n <= -9e+60) {
		tmp = t_0;
	} else if (n <= -6e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.5e-222) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else if (n <= 1.85e-39) {
		tmp = i * ((n * 100.0) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))))
	tmp = 0
	if n <= -9e+60:
		tmp = t_0
	elif n <= -6e-218:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.5e-222:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	elif n <= 1.85e-39:
		tmp = i * ((n * 100.0) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(0.041666666666666664 * Float64(i * Float64(i * i)))))
	tmp = 0.0
	if (n <= -9e+60)
		tmp = t_0;
	elseif (n <= -6e-218)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.5e-222)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	elseif (n <= 1.85e-39)
		tmp = Float64(i * Float64(Float64(n * 100.0) / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * 100.0) * (1.0 + (0.041666666666666664 * (i * (i * i))));
	tmp = 0.0;
	if (n <= -9e+60)
		tmp = t_0;
	elseif (n <= -6e-218)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.5e-222)
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	elseif (n <= 1.85e-39)
		tmp = i * ((n * 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), $MachinePrecision] * N[(1.0 + N[(0.041666666666666664 * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9e+60], t$95$0, If[LessEqual[n, -6e-218], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-222], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-39], N[(i * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.00000000000000026e60 or 1.85000000000000007e-39 < n

   1. Initial program 24.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 66.8%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 67.0%

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

   10. Simplified 67.0%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]

   11. Taylor expanded in i around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {i}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({i}^{3}\right)}\right)\right)\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.3%

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

   13. Simplified 66.3%

       \[\leadsto \left(100 \cdot n\right) \cdot \left(1 + \color{blue}{0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right) \]

   if -9.00000000000000026e60 < n < -5.9999999999999997e-218

   1. Initial program 31.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 64.4%

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

   if -5.9999999999999997e-218 < n < 5.50000000000000003e-222

   1. Initial program 76.6%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. sub-div N/A

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

      5. frac-sub N/A

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

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

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

   4. Applied egg-rr 45.8%

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

   5. Taylor expanded in i around 0

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

      1. /-lowering-/.f64 80.7%

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

   7. Simplified 80.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{n}}}{\frac{\frac{1}{n}}{n}} \]

   if 5.50000000000000003e-222 < n < 1.85000000000000007e-39

   1. Initial program 8.3%

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

   3. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 3.0%

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

   5. Simplified 3.0%

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

   6. Taylor expanded in i around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

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

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

      13. /-lowering-/.f64 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in i around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-lowering-*.f64 69.4%

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

   11. Simplified 69.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + 0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + 0.041666666666666664 \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.6e+172)
   (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))
   (if (<= n 3.8e-26)
     (/ 100.0 (+ (/ 1.0 n) (* i (+ (/ 0.5 (* n n)) (/ -0.5 n)))))
     (+
      (* n 100.0)
      (*
       i
       (*
        n
        (+
         50.0
         (*
          (* i 100.0)
          (+ 0.16666666666666666 (* i 0.041666666666666664))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.6e+172) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.6d+172)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    else if (n <= 3.8d-26) then
        tmp = 100.0d0 / ((1.0d0 / n) + (i * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    else
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + ((i * 100.0d0) * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.6e+172) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.6e+172:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	elif n <= 3.8e-26:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))))
	else:
		tmp = (n * 100.0) + (i * (n * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.6e+172)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	elseif (n <= 3.8e-26)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(Float64(i * 100.0) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.6e+172)
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	elseif (n <= 3.8e-26)
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	else
		tmp = (n * 100.0) + (i * (n * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.6e+172], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-26], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.6000000000000002e172

   1. Initial program 16.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 67.9%

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

   10. Simplified 67.9%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)} \]

   if -9.6000000000000002e172 < n < 3.80000000000000015e-26

   1. Initial program 32.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

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

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

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

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

      11. metadata-eval 32.4%

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

   4. Applied egg-rr 32.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 65.6%

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

   7. Simplified 65.6%

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

   if 3.80000000000000015e-26 < n

   1. Initial program 29.5%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 77.0%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 77.0%

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

   8. Simplified 77.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+171}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e+171)
   (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))
   (if (<= n 3.8e-26)
     (/ 100.0 (+ (/ 1.0 n) (* i (+ (/ 0.5 (* n n)) (/ -0.5 n)))))
     (*
      n
      (+
       100.0
       (*
        i
        (+
         50.0
         (*
          (* i 100.0)
          (+ 0.16666666666666666 (* i 0.041666666666666664))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e+171) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.5d+171)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    else if (n <= 3.8d-26) then
        tmp = 100.0d0 / ((1.0d0 / n) + (i * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + ((i * 100.0d0) * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.5e+171) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else if (n <= 3.8e-26) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	} else {
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.5e+171:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	elif n <= 3.8e-26:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))))
	else:
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e+171)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	elseif (n <= 3.8e-26)
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(Float64(i * 100.0) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.5e+171)
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	elseif (n <= 3.8e-26)
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 / (n * n)) + (-0.5 / n))));
	else
		tmp = n * (100.0 + (i * (50.0 + ((i * 100.0) * (0.16666666666666666 + (i * 0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e+171], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-26], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5000000000000002e171

   1. Initial program 16.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 67.9%

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

   10. Simplified 67.9%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)} \]

   if -2.5000000000000002e171 < n < 3.80000000000000015e-26

   1. Initial program 32.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

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

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

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

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

      11. metadata-eval 32.4%

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

   4. Applied egg-rr 32.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 65.6%

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

   7. Simplified 65.6%

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

   if 3.80000000000000015e-26 < n

   1. Initial program 29.5%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 77.0%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 77.0%

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

   8. Simplified 77.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+171}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \left(i \cdot 100\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 160:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.82)
   (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
   (if (<= i 160.0)
     (* 100.0 (+ n (* i (* n (+ 0.5 (/ -0.5 n))))))
     (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 160.0) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= (-1.82d0)) then
        tmp = 100.0d0 * ((1.0d0 + (-1.0d0)) / (i / n))
    else if (i <= 160.0d0) then
        tmp = 100.0d0 * (n + (i * (n * (0.5d0 + ((-0.5d0) / n)))))
    else
        tmp = 100.0d0 * ((1.0d0 / n) / ((1.0d0 / n) / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 160.0) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.82:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	elif i <= 160.0:
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))))
	else:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.82)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	elseif (i <= 160.0)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))));
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.82)
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	elseif (i <= 160.0)
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	else
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.82], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 160.0], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.82000000000000006

   1. Initial program 59.8%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 32.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]

   if -1.82000000000000006 < i < 160

   1. Initial program 7.9%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 82.3%

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

   5. Simplified 82.3%

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

   if 160 < i

   1. Initial program 43.3%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. sub-div N/A

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

      5. frac-sub N/A

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

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

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

   4. Applied egg-rr 34.7%

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

   5. Taylor expanded in i around 0

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

      1. /-lowering-/.f64 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 160:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 160:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
   (if (<= i 160.0)
     (+ (* n 100.0) (* 50.0 (* i n)))
     (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 160.0) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= (-2.0d0)) then
        tmp = 100.0d0 * ((1.0d0 + (-1.0d0)) / (i / n))
    else if (i <= 160.0d0) then
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    else
        tmp = 100.0d0 * ((1.0d0 / n) / ((1.0d0 / n) / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 160.0) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	elif i <= 160.0:
		tmp = (n * 100.0) + (50.0 * (i * n))
	else:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	elseif (i <= 160.0)
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	elseif (i <= 160.0)
		tmp = (n * 100.0) + (50.0 * (i * n));
	else
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 160.0], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 59.8%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 32.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]

   if -2 < i < 160

   1. Initial program 7.9%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.4%

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

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \color{blue}{\left(\frac{-25 \cdot {i}^{2} + 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)}{{n}^{2}}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\left(-25 \cdot {i}^{2} + 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \color{blue}{\left({n}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-25 \cdot {i}^{2}\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({\color{blue}{n}}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({i}^{2} \cdot -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \left(\left(100 \cdot i\right) \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\left(100 \cdot i\right), \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \left(i \cdot \frac{11}{24}\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \left(n \cdot \color{blue}{n}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 75.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \color{blue}{n}\right)\right)\right)\right)\right) \]

   8. Simplified 75.2%

      \[\leadsto 100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right) + \color{blue}{\frac{\left(i \cdot i\right) \cdot -25 + \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.3333333333333333 + i \cdot 0.4583333333333333\right)\right)}{n \cdot n}}\right) \]

   9. Taylor expanded in n around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 82.2%

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

   11. Simplified 82.2%

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

   if 160 < i

   1. Initial program 43.3%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-sub N/A

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

      4. sub-div N/A

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

      5. frac-sub N/A

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

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

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

   4. Applied egg-rr 34.7%

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

   5. Taylor expanded in i around 0

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

      1. /-lowering-/.f64 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 160:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+21}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
   (if (<= i 7e+21)
     (+ (* n 100.0) (* 50.0 (* i n)))
     (* 4.166666666666667 (* n (* i (* i i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 7e+21) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2.0d0)) then
        tmp = 100.0d0 * ((1.0d0 + (-1.0d0)) / (i / n))
    else if (i <= 7d+21) then
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    else
        tmp = 4.166666666666667d0 * (n * (i * (i * i)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 7e+21) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	elif i <= 7e+21:
		tmp = (n * 100.0) + (50.0 * (i * n))
	else:
		tmp = 4.166666666666667 * (n * (i * (i * i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	elseif (i <= 7e+21)
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	else
		tmp = Float64(4.166666666666667 * Float64(n * Float64(i * Float64(i * i))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	elseif (i <= 7e+21)
		tmp = (n * 100.0) + (50.0 * (i * n));
	else
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7e+21], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.166666666666667 * N[(n * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 59.8%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 32.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]

   if -2 < i < 7e21

   1. Initial program 9.2%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 67.9%

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

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \color{blue}{\left(\frac{-25 \cdot {i}^{2} + 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)}{{n}^{2}}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\left(-25 \cdot {i}^{2} + 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \color{blue}{\left({n}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-25 \cdot {i}^{2}\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({\color{blue}{n}}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({i}^{2} \cdot -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \left(100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \left(\left(100 \cdot i\right) \cdot \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\left(100 \cdot i\right), \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \left(n \cdot \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \left(\frac{1}{3} + \frac{11}{24} \cdot i\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{11}{24} \cdot i\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \left(i \cdot \frac{11}{24}\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \left({n}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \left(n \cdot \color{blue}{n}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 73.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -25\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(i, \frac{11}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \color{blue}{n}\right)\right)\right)\right)\right) \]

   8. Simplified 73.6%

      \[\leadsto 100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right) + \color{blue}{\frac{\left(i \cdot i\right) \cdot -25 + \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.3333333333333333 + i \cdot 0.4583333333333333\right)\right)}{n \cdot n}}\right) \]

   9. Taylor expanded in n around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 80.5%

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

   11. Simplified 80.5%

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

   if 7e21 < i

   1. Initial program 42.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. expm1-define N/A

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

      7. expm1-lowering-expm1.f64 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in i around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 50.7%

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

   10. Simplified 50.7%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]

   11. Taylor expanded in i around inf

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

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

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

       2. *-commutative N/A

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

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

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

       4. cube-mult N/A

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

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \left(i \cdot {i}^{\color{blue}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left({i}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

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

       8. *-lowering-*.f64 50.7%

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

   13. Simplified 50.7%

       \[\leadsto \color{blue}{4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+21}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.8:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.8)
   (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
   (if (<= i 7e+21)
     (* n (+ 100.0 (* i 50.0)))
     (* 4.166666666666667 (* n (* i (* i i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.8) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 7e+21) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.8d0)) then
        tmp = 100.0d0 * ((1.0d0 + (-1.0d0)) / (i / n))
    else if (i <= 7d+21) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 4.166666666666667d0 * (n * (i * (i * i)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.8) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (i <= 7e+21) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.8:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	elif i <= 7e+21:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 4.166666666666667 * (n * (i * (i * i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.8)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	elseif (i <= 7e+21)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(4.166666666666667 * Float64(n * Float64(i * Float64(i * i))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.8)
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	elseif (i <= 7e+21)
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.8], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7e+21], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.166666666666667 * N[(n * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.80000000000000004

   1. Initial program 59.8%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 32.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]

   if -1.80000000000000004 < i < 7e21

   1. Initial program 9.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 73.2%

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

   4. Applied egg-rr 73.2%

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

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. 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-in 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right) \]

      7. *-lowering-*.f64 80.5%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right) \]

   10. Simplified 80.5%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]

   if 7e21 < i

   1. Initial program 42.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 77.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 77.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 65.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 65.4%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 50.7%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]

   11. Taylor expanded in i around inf

       \[\leadsto \color{blue}{\frac{25}{6} \cdot \left({i}^{3} \cdot n\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \color{blue}{\left({i}^{3} \cdot n\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \left(n \cdot \color{blue}{{i}^{3}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \color{blue}{\left({i}^{3}\right)}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \left(i \cdot \color{blue}{\left(i \cdot i\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \left(i \cdot {i}^{\color{blue}{2}}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left({i}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(i \cdot \color{blue}{i}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 50.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right)\right) \]

   13. Simplified 50.7%

       \[\leadsto \color{blue}{4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.8:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.45:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.45)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 7e+21)
     (* n (+ 100.0 (* i 50.0)))
     (* 4.166666666666667 (* n (* i (* i i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.45) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 7e+21) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.45d0)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 7d+21) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 4.166666666666667d0 * (n * (i * (i * i)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.45) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 7e+21) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.45:
		tmp = 100.0 * (i / (i / n))
	elif i <= 7e+21:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 4.166666666666667 * (n * (i * (i * i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.45)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 7e+21)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(4.166666666666667 * Float64(n * Float64(i * Float64(i * i))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.45)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 7e+21)
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 4.166666666666667 * (n * (i * (i * i)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.45], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7e+21], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.166666666666667 * N[(n * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.44999999999999996

   1. Initial program 59.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 24.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -1.44999999999999996 < i < 7e21

   1. Initial program 9.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 73.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 73.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. 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-in 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right) \]

      7. *-lowering-*.f64 80.5%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right) \]

   10. Simplified 80.5%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]

   if 7e21 < i

   1. Initial program 42.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 77.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 77.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 65.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 65.4%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 50.7%

       \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)} \]

   11. Taylor expanded in i around inf

       \[\leadsto \color{blue}{\frac{25}{6} \cdot \left({i}^{3} \cdot n\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \color{blue}{\left({i}^{3} \cdot n\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \left(n \cdot \color{blue}{{i}^{3}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \color{blue}{\left({i}^{3}\right)}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \left(i \cdot \color{blue}{\left(i \cdot i\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \left(i \cdot {i}^{\color{blue}{2}}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left({i}^{2}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(i \cdot \color{blue}{i}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 50.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{25}{6}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right)\right) \]

   13. Simplified 50.7%

       \[\leadsto \color{blue}{4.166666666666667 \cdot \left(n \cdot \left(i \cdot \left(i \cdot i\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2e+72)
   (* (* i n) (/ 100.0 i))
   (if (<= n 3.8e-26) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2e+72) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 3.8e-26) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2d+72)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= 3.8d-26) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2e+72) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 3.8e-26) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2e+72:
		tmp = (i * n) * (100.0 / i)
	elif n <= 3.8e-26:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2e+72)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= 3.8e-26)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2e+72)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= 3.8e-26)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2e+72], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-26], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.99999999999999989e72

   1. Initial program 19.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{100}{\color{blue}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\left(\frac{i}{n}\right), \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      11. metadata-eval 19.2%

          \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), -1\right)\right)\right) \]

   4. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\color{blue}{\left(1 + i\right)}, -1\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\left(i + 1\right), -1\right)\right)\right) \]

      2. +-lowering-+.f64 3.9%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(i, 1\right), -1\right)\right)\right) \]

   7. Simplified 3.9%

      \[\leadsto \frac{100}{\frac{\frac{i}{n}}{\color{blue}{\left(i + 1\right)} + -1}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{100}{\frac{i}{\color{blue}{\left(\left(i + 1\right) + -1\right) \cdot n}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(\left(\left(i + 1\right) + -1\right) \cdot n\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{100}{i}\right), \color{blue}{\left(\left(\left(i + 1\right) + -1\right) \cdot n\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(100, i\right), \left(\color{blue}{\left(\left(i + 1\right) + -1\right)} \cdot n\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(100, i\right), \left(\left(i + \left(1 + -1\right)\right) \cdot n\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(100, i\right), \left(\left(i + 0\right) \cdot n\right)\right) \]

      7. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(100, i\right), \left(i \cdot n\right)\right) \]

      8. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(100, i\right), \mathsf{*.f64}\left(i, \color{blue}{n}\right)\right) \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{\frac{100}{i} \cdot \left(i \cdot n\right)} \]

   if -1.99999999999999989e72 < n < 3.80000000000000015e-26

   1. Initial program 35.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 59.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 3.80000000000000015e-26 < n

   1. Initial program 29.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 91.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 91.6%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. 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-in 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right) \]

      7. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right) \]

   10. Simplified 56.0%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -9.5e+60) t_0 (if (<= n 3.8e-26) (* 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 <= -9.5e+60) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		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 <= (-9.5d+60)) then
        tmp = t_0
    else if (n <= 3.8d-26) 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 <= -9.5e+60) {
		tmp = t_0;
	} else if (n <= 3.8e-26) {
		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 <= -9.5e+60:
		tmp = t_0
	elif n <= 3.8e-26:
		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 <= -9.5e+60)
		tmp = t_0;
	elseif (n <= 3.8e-26)
		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 <= -9.5e+60)
		tmp = t_0;
	elseif (n <= 3.8e-26)
		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, -9.5e+60], t$95$0, If[LessEqual[n, 3.8e-26], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.49999999999999988e60 or 3.80000000000000015e-26 < n

   1. Initial program 24.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 66.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 66.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\frac{\color{blue}{e^{i} - 1}}{i}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right) \]

      7. expm1-lowering-expm1.f64 92.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right) \]

   7. Simplified 92.0%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. 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-in 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right) \]

      7. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right) \]

   10. Simplified 56.8%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]

   if -9.49999999999999988e60 < n < 3.80000000000000015e-26

   1. Initial program 34.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 60.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -5 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= i -5e+49) t_0 (if (<= i 1.4e-46) (* n 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (i <= -5e+49) {
		tmp = t_0;
	} else if (i <= 1.4e-46) {
		tmp = n * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (i <= (-5d+49)) then
        tmp = t_0
    else if (i <= 1.4d-46) then
        tmp = n * 100.0d0
    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 <= -5e+49) {
		tmp = t_0;
	} else if (i <= 1.4e-46) {
		tmp = n * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if i <= -5e+49:
		tmp = t_0
	elif i <= 1.4e-46:
		tmp = n * 100.0
	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 <= -5e+49)
		tmp = t_0;
	elseif (i <= 1.4e-46)
		tmp = Float64(n * 100.0);
	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 <= -5e+49)
		tmp = t_0;
	elseif (i <= 1.4e-46)
		tmp = n * 100.0;
	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, -5e+49], t$95$0, If[LessEqual[i, 1.4e-46], N[(n * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -5.0000000000000004e49 or 1.3999999999999999e-46 < i

   1. Initial program 46.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 25.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -5.0000000000000004e49 < i < 1.3999999999999999e-46

   1. Initial program 10.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 80.2%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+49}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.6% 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 29.0%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 44.9%

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

5. Simplified 44.9%

   \[\leadsto \color{blue}{100 \cdot n} \]

6. Final simplification 44.9%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer Target 1: 34.0% 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 2024138 
(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))))