Compound Interest

Average Error: 47.9 → 10.0

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {n}^{3} \cdot \log n\\ t_1 := \log n \cdot {n}^{4}\\ t_2 := {\log n}^{3}\\ t_3 := {\log n}^{2}\\ t_4 := t_3 \cdot {n}^{4}\\ t_5 := {\log i}^{2}\\ t_6 := {\log i}^{3}\\ \mathbf{if}\;i \leq -2.6748076836181006 \cdot 10^{-23}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.009094777292108976:\\ \;\;\;\;n \cdot 100 + n \cdot \left(\mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right) - 50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{fma}\left(100, \frac{{n}^{3} \cdot \log i}{i \cdot i}, \mathsf{fma}\left(50, \frac{\log i \cdot \left({n}^{3} \cdot t_3\right)}{i}, \mathsf{fma}\left(50, \frac{t_1}{{i}^{3}}, \mathsf{fma}\left(50, \frac{{n}^{4}}{{i}^{3}}, \mathsf{fma}\left(33.333333333333336, \frac{{n}^{4}}{{i}^{4}}, \mathsf{fma}\left(100, \frac{n \cdot \log i}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log i}^{4}}{i}, \mathsf{fma}\left(50, \frac{\left(n \cdot n\right) \cdot t_5}{i}, \mathsf{fma}\left(50, \frac{t_3 \cdot \left(n \cdot n\right)}{i}, \mathsf{fma}\left(25, \frac{t_5 \cdot t_4}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log n}^{4}}{i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot t_6}{i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot t_5}{i \cdot i}, \mathsf{fma}\left(50, \frac{t_4}{i \cdot i}, 100 \cdot \left(\frac{n}{i} \cdot \frac{n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(16.666666666666668, \frac{\log i \cdot \left({n}^{4} \cdot t_2\right)}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot t_1}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{t_1 \cdot t_6}{i}, \mathsf{fma}\left(50, {\left(\frac{n}{i}\right)}^{3}, \mathsf{fma}\left(50, \frac{t_5 \cdot t_0}{i}, \mathsf{fma}\left(50, \frac{\log i \cdot {n}^{4}}{{i}^{3}}, \mathsf{fma}\left(100, \frac{t_0}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot t_2}{i}, 100 \cdot \left(\frac{\left(n \cdot n\right) \cdot \left(\log i \cdot \log n\right)}{i} + \frac{n \cdot \log n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (pow n 3.0) (log n)))
        (t_1 (* (log n) (pow n 4.0)))
        (t_2 (pow (log n) 3.0))
        (t_3 (pow (log n) 2.0))
        (t_4 (* t_3 (pow n 4.0)))
        (t_5 (pow (log i) 2.0))
        (t_6 (pow (log i) 3.0)))
   (if (<= i -2.6748076836181006e-23)
     (* n (/ (fma 100.0 (exp (* n (log1p (/ i n)))) -100.0) i))
     (if (<= i 0.009094777292108976)
       (+
        (* n 100.0)
        (*
         n
         (-
          (fma
           50.0
           i
           (fma
            16.666666666666668
            (* i i)
            (* 33.333333333333336 (* (/ i n) (/ i n)))))
          (* 50.0 (+ (/ i n) (/ (* i i) n))))))
       (*
        n
        (-
         (fma
          100.0
          (/ (* (pow n 3.0) (log i)) (* i i))
          (fma
           50.0
           (/ (* (log i) (* (pow n 3.0) t_3)) i)
           (fma
            50.0
            (/ t_1 (pow i 3.0))
            (fma
             50.0
             (/ (pow n 4.0) (pow i 3.0))
             (fma
              33.333333333333336
              (/ (pow n 4.0) (pow i 4.0))
              (fma
               100.0
               (/ (* n (log i)) i)
               (fma
                4.166666666666667
                (/ (* (pow n 4.0) (pow (log i) 4.0)) i)
                (fma
                 50.0
                 (/ (* (* n n) t_5) i)
                 (fma
                  50.0
                  (/ (* t_3 (* n n)) i)
                  (fma
                   25.0
                   (/ (* t_5 t_4) i)
                   (fma
                    4.166666666666667
                    (/ (* (pow n 4.0) (pow (log n) 4.0)) i)
                    (fma
                     16.666666666666668
                     (/ (* (pow n 3.0) t_6) i)
                     (fma
                      50.0
                      (/ (* (pow n 4.0) t_5) (* i i))
                      (fma
                       50.0
                       (/ t_4 (* i i))
                       (* 100.0 (* (/ n i) (/ n i)))))))))))))))))
         (fma
          16.666666666666668
          (/ (* (log i) (* (pow n 4.0) t_2)) i)
          (fma
           100.0
           (/ (* (log i) t_1) (* i i))
           (fma
            16.666666666666668
            (/ (* t_1 t_6) i)
            (fma
             50.0
             (pow (/ n i) 3.0)
             (fma
              50.0
              (/ (* t_5 t_0) i)
              (fma
               50.0
               (/ (* (log i) (pow n 4.0)) (pow i 3.0))
               (fma
                100.0
                (/ t_0 (* i i))
                (fma
                 16.666666666666668
                 (/ (* (pow n 3.0) t_2) i)
                 (*
                  100.0
                  (+
                   (/ (* (* n n) (* (log i) (log n))) i)
                   (/ (* n (log n)) i)))))))))))))))))

C

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

↓

double code(double i, double n) {
	double t_0 = pow(n, 3.0) * log(n);
	double t_1 = log(n) * pow(n, 4.0);
	double t_2 = pow(log(n), 3.0);
	double t_3 = pow(log(n), 2.0);
	double t_4 = t_3 * pow(n, 4.0);
	double t_5 = pow(log(i), 2.0);
	double t_6 = pow(log(i), 3.0);
	double tmp;
	if (i <= -2.6748076836181006e-23) {
		tmp = n * (fma(100.0, exp((n * log1p((i / n)))), -100.0) / i);
	} else if (i <= 0.009094777292108976) {
		tmp = (n * 100.0) + (n * (fma(50.0, i, fma(16.666666666666668, (i * i), (33.333333333333336 * ((i / n) * (i / n))))) - (50.0 * ((i / n) + ((i * i) / n)))));
	} else {
		tmp = n * (fma(100.0, ((pow(n, 3.0) * log(i)) / (i * i)), fma(50.0, ((log(i) * (pow(n, 3.0) * t_3)) / i), fma(50.0, (t_1 / pow(i, 3.0)), fma(50.0, (pow(n, 4.0) / pow(i, 3.0)), fma(33.333333333333336, (pow(n, 4.0) / pow(i, 4.0)), fma(100.0, ((n * log(i)) / i), fma(4.166666666666667, ((pow(n, 4.0) * pow(log(i), 4.0)) / i), fma(50.0, (((n * n) * t_5) / i), fma(50.0, ((t_3 * (n * n)) / i), fma(25.0, ((t_5 * t_4) / i), fma(4.166666666666667, ((pow(n, 4.0) * pow(log(n), 4.0)) / i), fma(16.666666666666668, ((pow(n, 3.0) * t_6) / i), fma(50.0, ((pow(n, 4.0) * t_5) / (i * i)), fma(50.0, (t_4 / (i * i)), (100.0 * ((n / i) * (n / i))))))))))))))))) - fma(16.666666666666668, ((log(i) * (pow(n, 4.0) * t_2)) / i), fma(100.0, ((log(i) * t_1) / (i * i)), fma(16.666666666666668, ((t_1 * t_6) / i), fma(50.0, pow((n / i), 3.0), fma(50.0, ((t_5 * t_0) / i), fma(50.0, ((log(i) * pow(n, 4.0)) / pow(i, 3.0)), fma(100.0, (t_0 / (i * i)), fma(16.666666666666668, ((pow(n, 3.0) * t_2) / i), (100.0 * ((((n * n) * (log(i) * log(n))) / i) + ((n * log(n)) / i))))))))))));
	}
	return tmp;
}

Julia

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

↓

function code(i, n)
	t_0 = Float64((n ^ 3.0) * log(n))
	t_1 = Float64(log(n) * (n ^ 4.0))
	t_2 = log(n) ^ 3.0
	t_3 = log(n) ^ 2.0
	t_4 = Float64(t_3 * (n ^ 4.0))
	t_5 = log(i) ^ 2.0
	t_6 = log(i) ^ 3.0
	tmp = 0.0
	if (i <= -2.6748076836181006e-23)
		tmp = Float64(n * Float64(fma(100.0, exp(Float64(n * log1p(Float64(i / n)))), -100.0) / i));
	elseif (i <= 0.009094777292108976)
		tmp = Float64(Float64(n * 100.0) + Float64(n * Float64(fma(50.0, i, fma(16.666666666666668, Float64(i * i), Float64(33.333333333333336 * Float64(Float64(i / n) * Float64(i / n))))) - Float64(50.0 * Float64(Float64(i / n) + Float64(Float64(i * i) / n))))));
	else
		tmp = Float64(n * Float64(fma(100.0, Float64(Float64((n ^ 3.0) * log(i)) / Float64(i * i)), fma(50.0, Float64(Float64(log(i) * Float64((n ^ 3.0) * t_3)) / i), fma(50.0, Float64(t_1 / (i ^ 3.0)), fma(50.0, Float64((n ^ 4.0) / (i ^ 3.0)), fma(33.333333333333336, Float64((n ^ 4.0) / (i ^ 4.0)), fma(100.0, Float64(Float64(n * log(i)) / i), fma(4.166666666666667, Float64(Float64((n ^ 4.0) * (log(i) ^ 4.0)) / i), fma(50.0, Float64(Float64(Float64(n * n) * t_5) / i), fma(50.0, Float64(Float64(t_3 * Float64(n * n)) / i), fma(25.0, Float64(Float64(t_5 * t_4) / i), fma(4.166666666666667, Float64(Float64((n ^ 4.0) * (log(n) ^ 4.0)) / i), fma(16.666666666666668, Float64(Float64((n ^ 3.0) * t_6) / i), fma(50.0, Float64(Float64((n ^ 4.0) * t_5) / Float64(i * i)), fma(50.0, Float64(t_4 / Float64(i * i)), Float64(100.0 * Float64(Float64(n / i) * Float64(n / i))))))))))))))))) - fma(16.666666666666668, Float64(Float64(log(i) * Float64((n ^ 4.0) * t_2)) / i), fma(100.0, Float64(Float64(log(i) * t_1) / Float64(i * i)), fma(16.666666666666668, Float64(Float64(t_1 * t_6) / i), fma(50.0, (Float64(n / i) ^ 3.0), fma(50.0, Float64(Float64(t_5 * t_0) / i), fma(50.0, Float64(Float64(log(i) * (n ^ 4.0)) / (i ^ 3.0)), fma(100.0, Float64(t_0 / Float64(i * i)), fma(16.666666666666668, Float64(Float64((n ^ 3.0) * t_2) / i), Float64(100.0 * Float64(Float64(Float64(Float64(n * n) * Float64(log(i) * log(n))) / i) + Float64(Float64(n * log(n)) / i)))))))))))));
	end
	return tmp
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]

↓

code[i_, n_] := Block[{t$95$0 = N[(N[Power[n, 3.0], $MachinePrecision] * N[Log[n], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[n], $MachinePrecision] * N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Log[n], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Log[n], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Log[i], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Log[i], $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[i, -2.6748076836181006e-23], N[(n * N[(N[(100.0 * N[Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 0.009094777292108976], N[(N[(n * 100.0), $MachinePrecision] + N[(n * N[(N[(50.0 * i + N[(16.666666666666668 * N[(i * i), $MachinePrecision] + N[(33.333333333333336 * N[(N[(i / n), $MachinePrecision] * N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(50.0 * N[(N[(i / n), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(N[(N[Power[n, 3.0], $MachinePrecision] * N[Log[i], $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(N[(N[Log[i], $MachinePrecision] * N[(N[Power[n, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[(t$95$1 / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(N[Power[n, 4.0], $MachinePrecision] / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision] + N[(33.333333333333336 * N[(N[Power[n, 4.0], $MachinePrecision] / N[Power[i, 4.0], $MachinePrecision]), $MachinePrecision] + N[(100.0 * N[(N[(n * N[Log[i], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(4.166666666666667 * N[(N[(N[Power[n, 4.0], $MachinePrecision] * N[Power[N[Log[i], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[(N[(N[(n * n), $MachinePrecision] * t$95$5), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[(N[(t$95$3 * N[(n * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(25.0 * N[(N[(t$95$5 * t$95$4), $MachinePrecision] / i), $MachinePrecision] + N[(4.166666666666667 * N[(N[(N[Power[n, 4.0], $MachinePrecision] * N[Power[N[Log[n], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(16.666666666666668 * N[(N[(N[Power[n, 3.0], $MachinePrecision] * t$95$6), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[(N[(N[Power[n, 4.0], $MachinePrecision] * t$95$5), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(t$95$4 / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(16.666666666666668 * N[(N[(N[Log[i], $MachinePrecision] * N[(N[Power[n, 4.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(100.0 * N[(N[(N[Log[i], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(16.666666666666668 * N[(N[(t$95$1 * t$95$6), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[Power[N[(n / i), $MachinePrecision], 3.0], $MachinePrecision] + N[(50.0 * N[(N[(t$95$5 * t$95$0), $MachinePrecision] / i), $MachinePrecision] + N[(50.0 * N[(N[(N[Log[i], $MachinePrecision] * N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision] + N[(100.0 * N[(t$95$0 / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(16.666666666666668 * N[(N[(N[Power[n, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision] / i), $MachinePrecision] + N[(100.0 * N[(N[(N[(N[(n * n), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(N[(n * N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Error

Bits error versus i

Bits error versus n

Target

Original	47.9
Target	47.4
Herbie	10.0

\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

1. Split input into 3 regimes

2. if i < -2.6748076836181006e-23

   1. Initial program 29.3

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

   2. Simplified 29.9

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

   3. Applied pow-to-exp_binary64 29.9

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

   4. Simplified 8.8

      \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{i} \]

   if -2.6748076836181006e-23 < i < 0.00909477729210897597

   1. Initial program 58.7

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

   2. Simplified 58.4

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

   3. Taylor expanded in i around 0 12.5

      \[\leadsto n \cdot \color{blue}{\left(\left(100 + \left(50 \cdot i + \left(16.666666666666668 \cdot {i}^{2} + 33.333333333333336 \cdot \frac{{i}^{2}}{{n}^{2}}\right)\right)\right) - \left(50 \cdot \frac{i}{n} + 50 \cdot \frac{{i}^{2}}{n}\right)\right)} \]

   4. Simplified 8.4

      \[\leadsto n \cdot \color{blue}{\left(\left(100 + \mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right)\right) - 50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)} \]

   5. Applied associate--l+_binary64 8.4

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(\mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right) - 50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)\right)} \]

   6. Applied distribute-lft-in_binary64 8.4

      \[\leadsto \color{blue}{n \cdot 100 + n \cdot \left(\mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right) - 50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)} \]

   if 0.00909477729210897597 < i

   1. Initial program 34.1

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

   2. Simplified 34.0

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

   3. Taylor expanded in n around 0 19.6

      \[\leadsto n \cdot \color{blue}{\left(\left(100 \cdot \frac{{n}^{3} \cdot \log i}{{i}^{2}} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot {\log n}^{2}\right)}{i} + \left(50 \cdot \frac{{n}^{4} \cdot \log n}{{i}^{3}} + \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(33.333333333333336 \cdot \frac{{n}^{4}}{{i}^{4}} + \left(100 \cdot \frac{n \cdot \log i}{i} + \left(4.166666666666667 \cdot \frac{{\log i}^{4} \cdot {n}^{4}}{i} + \left(50 \cdot \frac{{n}^{2} \cdot {\log i}^{2}}{i} + \left(50 \cdot \frac{{n}^{2} \cdot {\log n}^{2}}{i} + \left(25 \cdot \frac{{\log i}^{2} \cdot \left({n}^{4} \cdot {\log n}^{2}\right)}{i} + \left(4.166666666666667 \cdot \frac{{n}^{4} \cdot {\log n}^{4}}{i} + \left(16.666666666666668 \cdot \frac{{n}^{3} \cdot {\log i}^{3}}{i} + \left(50 \cdot \frac{{\log i}^{2} \cdot {n}^{4}}{{i}^{2}} + \left(50 \cdot \frac{{n}^{4} \cdot {\log n}^{2}}{{i}^{2}} + 100 \cdot \frac{{n}^{2}}{{i}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(16.666666666666668 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\log n}^{3}\right)}{i} + \left(100 \cdot \frac{\log i \cdot \left({n}^{4} \cdot \log n\right)}{{i}^{2}} + \left(16.666666666666668 \cdot \frac{{\log i}^{3} \cdot \left({n}^{4} \cdot \log n\right)}{i} + \left(50 \cdot \frac{{n}^{3}}{{i}^{3}} + \left(50 \cdot \frac{{n}^{3} \cdot \left({\log i}^{2} \cdot \log n\right)}{i} + \left(50 \cdot \frac{\log i \cdot {n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{3} \cdot \log n}{{i}^{2}} + \left(16.666666666666668 \cdot \frac{{n}^{3} \cdot {\log n}^{3}}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \left(\log i \cdot \log n\right)}{i} + 100 \cdot \frac{n \cdot \log n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

   4. Simplified 19.6

      \[\leadsto n \cdot \color{blue}{\left(\mathsf{fma}\left(100, \frac{{n}^{3} \cdot \log i}{i \cdot i}, \mathsf{fma}\left(50, \frac{\log i \cdot \left({n}^{3} \cdot {\log n}^{2}\right)}{i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot \log n}{{i}^{3}}, \mathsf{fma}\left(50, \frac{{n}^{4}}{{i}^{3}}, \mathsf{fma}\left(33.333333333333336, \frac{{n}^{4}}{{i}^{4}}, \mathsf{fma}\left(100, \frac{n \cdot \log i}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log i}^{4}}{i}, \mathsf{fma}\left(50, \frac{\left(n \cdot n\right) \cdot {\log i}^{2}}{i}, \mathsf{fma}\left(50, \frac{\left(n \cdot n\right) \cdot {\log n}^{2}}{i}, \mathsf{fma}\left(25, \frac{{\log i}^{2} \cdot \left({n}^{4} \cdot {\log n}^{2}\right)}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log n}^{4}}{i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot {\log i}^{3}}{i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot {\log i}^{2}}{i \cdot i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot {\log n}^{2}}{i \cdot i}, 100 \cdot \left(\frac{n}{i} \cdot \frac{n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(16.666666666666668, \frac{\log i \cdot \left({n}^{4} \cdot {\log n}^{3}\right)}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot \left({n}^{4} \cdot \log n\right)}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{{\log i}^{3} \cdot \left({n}^{4} \cdot \log n\right)}{i}, \mathsf{fma}\left(50, {\left(\frac{n}{i}\right)}^{3}, \mathsf{fma}\left(50, \frac{{\log i}^{2} \cdot \left({n}^{3} \cdot \log n\right)}{i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot \log i}{{i}^{3}}, \mathsf{fma}\left(100, \frac{{n}^{3} \cdot \log n}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot {\log n}^{3}}{i}, 100 \cdot \left(\frac{\left(n \cdot n\right) \cdot \left(\log i \cdot \log n\right)}{i} + \frac{n \cdot \log n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.6748076836181006 \cdot 10^{-23}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.009094777292108976:\\ \;\;\;\;n \cdot 100 + n \cdot \left(\mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right) - 50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{fma}\left(100, \frac{{n}^{3} \cdot \log i}{i \cdot i}, \mathsf{fma}\left(50, \frac{\log i \cdot \left({n}^{3} \cdot {\log n}^{2}\right)}{i}, \mathsf{fma}\left(50, \frac{\log n \cdot {n}^{4}}{{i}^{3}}, \mathsf{fma}\left(50, \frac{{n}^{4}}{{i}^{3}}, \mathsf{fma}\left(33.333333333333336, \frac{{n}^{4}}{{i}^{4}}, \mathsf{fma}\left(100, \frac{n \cdot \log i}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log i}^{4}}{i}, \mathsf{fma}\left(50, \frac{\left(n \cdot n\right) \cdot {\log i}^{2}}{i}, \mathsf{fma}\left(50, \frac{{\log n}^{2} \cdot \left(n \cdot n\right)}{i}, \mathsf{fma}\left(25, \frac{{\log i}^{2} \cdot \left({\log n}^{2} \cdot {n}^{4}\right)}{i}, \mathsf{fma}\left(4.166666666666667, \frac{{n}^{4} \cdot {\log n}^{4}}{i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot {\log i}^{3}}{i}, \mathsf{fma}\left(50, \frac{{n}^{4} \cdot {\log i}^{2}}{i \cdot i}, \mathsf{fma}\left(50, \frac{{\log n}^{2} \cdot {n}^{4}}{i \cdot i}, 100 \cdot \left(\frac{n}{i} \cdot \frac{n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(16.666666666666668, \frac{\log i \cdot \left({n}^{4} \cdot {\log n}^{3}\right)}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot \left(\log n \cdot {n}^{4}\right)}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{\left(\log n \cdot {n}^{4}\right) \cdot {\log i}^{3}}{i}, \mathsf{fma}\left(50, {\left(\frac{n}{i}\right)}^{3}, \mathsf{fma}\left(50, \frac{{\log i}^{2} \cdot \left({n}^{3} \cdot \log n\right)}{i}, \mathsf{fma}\left(50, \frac{\log i \cdot {n}^{4}}{{i}^{3}}, \mathsf{fma}\left(100, \frac{{n}^{3} \cdot \log n}{i \cdot i}, \mathsf{fma}\left(16.666666666666668, \frac{{n}^{3} \cdot {\log n}^{3}}{i}, 100 \cdot \left(\frac{\left(n \cdot n\right) \cdot \left(\log i \cdot \log n\right)}{i} + \frac{n \cdot \log n}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))