Compound Interest

Average Error: 47.7 → 12.0

Time: 19.2s

Precision: binary64

Cost: 40332

Math

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

↓

\[\begin{array}{l} t_0 := \log i - \log n\\ t_1 := \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -5.4 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot t_0 + n \cdot \left(n \cdot \mathsf{fma}\left(0.5, {t_0}^{2}, \frac{1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (log i) (log n))) (t_1 (/ (* n 100.0) (/ i (expm1 i)))))
   (if (<= n -5.4e-140)
     t_1
     (if (<= n -5e-310)
       (/
        (* 100.0 (expm1 (* n (- (log (/ -1.0 n)) (log (/ -1.0 i))))))
        (/ i n))
       (if (<= n 1.2e-92)
         (/
          (* 100.0 (+ (* n t_0) (* n (* n (fma 0.5 (pow t_0 2.0) (/ 1.0 i))))))
          (/ i n))
         t_1)))))

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 = log(i) - log(n);
	double t_1 = (n * 100.0) / (i / expm1(i));
	double tmp;
	if (n <= -5.4e-140) {
		tmp = t_1;
	} else if (n <= -5e-310) {
		tmp = (100.0 * expm1((n * (log((-1.0 / n)) - log((-1.0 / i)))))) / (i / n);
	} else if (n <= 1.2e-92) {
		tmp = (100.0 * ((n * t_0) + (n * (n * fma(0.5, pow(t_0, 2.0), (1.0 / i)))))) / (i / n);
	} else {
		tmp = t_1;
	}
	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(log(i) - log(n))
	t_1 = Float64(Float64(n * 100.0) / Float64(i / expm1(i)))
	tmp = 0.0
	if (n <= -5.4e-140)
		tmp = t_1;
	elseif (n <= -5e-310)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * Float64(log(Float64(-1.0 / n)) - log(Float64(-1.0 / i)))))) / Float64(i / n));
	elseif (n <= 1.2e-92)
		tmp = Float64(Float64(100.0 * Float64(Float64(n * t_0) + Float64(n * Float64(n * fma(0.5, (t_0 ^ 2.0), Float64(1.0 / i)))))) / Float64(i / n));
	else
		tmp = t_1;
	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[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.4e-140], t$95$1, If[LessEqual[n, -5e-310], N[(N[(100.0 * N[(Exp[N[(n * N[(N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-92], N[(N[(100.0 * N[(N[(n * t$95$0), $MachinePrecision] + N[(n * N[(n * N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Target

Original	47.7
Target	47.2
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if n < -5.4e-140 or 1.2000000000000001e-92 < n

   1. Initial program 51.8

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

   2. Simplified 51.6

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

      [Start] 51.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 51.6	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 51.6	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 51.6	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 51.6	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 51.6	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 51.6	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 51.6	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 48.5

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

   4. Simplified 9.9

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

      [Start] 48.5	\[ 100 \cdot \left(n \cdot \frac{e^{i} - 1}{i}\right) \]
      expm1-def [=>] 9.9	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]

   5. Applied egg-rr 10.0

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

   if -5.4e-140 < n < -4.999999999999985e-310

   1. Initial program 23.4

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

   2. Simplified 23.4

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

      [Start] 23.4	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 23.4	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 23.4	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 23.4	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around -inf 23.8

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

   4. Simplified 15.9

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

      [Start] 23.8	\[ \frac{100 \cdot \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1\right)}{\frac{i}{n}} \]
      expm1-def [=>] 15.9	\[ \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)\right)}}{\frac{i}{n}} \]
      +-commutative [=>] 15.9	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(\log \left(-\frac{1}{n}\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}\right)}{\frac{i}{n}} \]
      mul-1-neg [=>] 15.9	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(-\frac{1}{n}\right) + \color{blue}{\left(-\log \left(\frac{-1}{i}\right)\right)}\right)\right)}{\frac{i}{n}} \]
      unsub-neg [=>] 15.9	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(\log \left(-\frac{1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)}\right)}{\frac{i}{n}} \]
      distribute-neg-frac [=>] 15.9	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \color{blue}{\left(\frac{-1}{n}\right)} - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]
      metadata-eval [=>] 15.9	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{\color{blue}{-1}}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]

   if -4.999999999999985e-310 < n < 1.2000000000000001e-92

   1. Initial program 46.3

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

   2. Simplified 46.3

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

      [Start] 46.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 46.3	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 46.3	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 46.3	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in n around 0 19.9

      \[\leadsto \frac{\color{blue}{100 \cdot \left({n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right) + 100 \cdot \left(n \cdot \left(-1 \cdot \log n + \log i\right)\right)}}{\frac{i}{n}} \]

   4. Simplified 19.7

      \[\leadsto \frac{\color{blue}{100 \cdot \left(n \cdot \left(\log i - \log n\right) + n \cdot \left(n \cdot \mathsf{fma}\left(0.5, {\left(\log i - \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}}{\frac{i}{n}} \]
      Proof

      [Start] 19.9	\[ \frac{100 \cdot \left({n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right) + 100 \cdot \left(n \cdot \left(-1 \cdot \log n + \log i\right)\right)}{\frac{i}{n}} \]
      distribute-lft-out [=>] 19.9	\[ \frac{\color{blue}{100 \cdot \left({n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right) + n \cdot \left(-1 \cdot \log n + \log i\right)\right)}}{\frac{i}{n}} \]
      +-commutative [=>] 19.9	\[ \frac{100 \cdot \color{blue}{\left(n \cdot \left(-1 \cdot \log n + \log i\right) + {n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}}{\frac{i}{n}} \]
      +-commutative [=>] 19.9	\[ \frac{100 \cdot \left(n \cdot \color{blue}{\left(\log i + -1 \cdot \log n\right)} + {n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}{\frac{i}{n}} \]
      mul-1-neg [=>] 19.9	\[ \frac{100 \cdot \left(n \cdot \left(\log i + \color{blue}{\left(-\log n\right)}\right) + {n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}{\frac{i}{n}} \]
      unsub-neg [=>] 19.9	\[ \frac{100 \cdot \left(n \cdot \color{blue}{\left(\log i - \log n\right)} + {n}^{2} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}{\frac{i}{n}} \]
      unpow2 [=>] 19.9	\[ \frac{100 \cdot \left(n \cdot \left(\log i - \log n\right) + \color{blue}{\left(n \cdot n\right)} \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}{\frac{i}{n}} \]
      associate-*l* [=>] 19.7	\[ \frac{100 \cdot \left(n \cdot \left(\log i - \log n\right) + \color{blue}{n \cdot \left(n \cdot \left(0.5 \cdot {\left(-1 \cdot \log n + \log i\right)}^{2} + \frac{1}{i}\right)\right)}\right)}{\frac{i}{n}} \]
      fma-def [=>] 19.7	\[ \frac{100 \cdot \left(n \cdot \left(\log i - \log n\right) + n \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(0.5, {\left(-1 \cdot \log n + \log i\right)}^{2}, \frac{1}{i}\right)}\right)\right)}{\frac{i}{n}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\log i - \log n\right) + n \cdot \left(n \cdot \mathsf{fma}\left(0.5, {\left(\log i - \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.7
Cost	7628

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\frac{t_0}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t_0}\\ \end{array} \]

Alternative 2
Error	12.3
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -4.8 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 500:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.4
Cost	7244

\[\begin{array}{l} t_0 := \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -2.95 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 40:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.4
Cost	7244

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{1}{\frac{t_0}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 500:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t_0}\\ \end{array} \]

Alternative 5
Error	19.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-220} \lor \neg \left(n \leq 1.9 \cdot 10^{-202}\right):\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 6
Error	19.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-219} \lor \neg \left(n \leq 2 \cdot 10^{-202}\right):\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 7
Error	23.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4 \lor \neg \left(i \leq 0.33\right):\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8
Error	23.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4 \lor \neg \left(i \leq 0.33\right):\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]

Alternative 9
Error	21.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-142} \lor \neg \left(n \leq 1.42 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{1}{\frac{\frac{i}{n}}{100}}\\ \end{array} \]

Alternative 10
Error	19.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-222} \lor \neg \left(n \leq 2.4 \cdot 10^{-202}\right):\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 11
Error	22.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.33:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{1}{\frac{\frac{i}{n}}{100}}\\ \end{array} \]

Alternative 12
Error	23.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4 \lor \neg \left(i \leq 0.33\right):\\ \;\;\;\;i \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13
Error	23.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4 \lor \neg \left(i \leq 0.33\right):\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 15
Error	28.4
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023039 
(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))))