Compound Interest

Average Accuracy: 25.7% → 98.4%

Time: 21.6s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-183}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, i \cdot -0.5\right)}\\ \end{array} \]

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 (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 4e-183)
     (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
     (if (<= t_0 5e+15)
       (* t_0 100.0)
       (* 100.0 (/ n (+ 1.0 (fma (* i i) 0.08333333333333333 (* i -0.5)))))))))

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((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 4e-183) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_0 <= 5e+15) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + fma((i * i), 0.08333333333333333, (i * -0.5))));
	}
	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(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 4e-183)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_0 <= 5e+15)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + fma(Float64(i * i), 0.08333333333333333, Float64(i * -0.5)))));
	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[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-183], N[(100.0 / N[(N[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+15], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * i), $MachinePrecision] * 0.08333333333333333 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	25.7%
Target	25.5%
Herbie	98.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 4.00000000000000002e-183

   1. Initial program 28.9%

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

   2. Applied egg-rr 98.2%

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

      [Start] 28.9	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      clear-num [=>] 28.9	\[ 100 \cdot \color{blue}{\frac{1}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]
      un-div-inv [=>] 28.9	\[ \color{blue}{\frac{100}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]
      pow-to-exp [=>] 28.1	\[ \frac{100}{\frac{\frac{i}{n}}{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}} \]
      expm1-def [=>] 39.9	\[ \frac{100}{\frac{\frac{i}{n}}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}} \]
      *-commutative [=>] 39.9	\[ \frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}} \]
      log1p-udef [<=] 98.2	\[ \frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}} \]

   if 4.00000000000000002e-183 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5e15

   1. Initial program 95.9%

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

   if 5e15 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.1%

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

   2. Taylor expanded in n around inf 1.6%

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

   3. Simplified 76.7%

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

      [Start] 1.6	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      *-commutative [=>] 1.6	\[ \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]
      associate-/l* [=>] 1.6	\[ \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]
      expm1-def [=>] 76.7	\[ \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]

   5. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)} \cdot 100 \]
      *-commutative [=>] 99.9	\[ \frac{n}{1 + \left(\color{blue}{{i}^{2} \cdot 0.08333333333333333} + -0.5 \cdot i\right)} \cdot 100 \]
      fma-def [=>] 99.9	\[ \frac{n}{1 + \color{blue}{\mathsf{fma}\left({i}^{2}, 0.08333333333333333, -0.5 \cdot i\right)}} \cdot 100 \]
      unpow2 [=>] 99.9	\[ \frac{n}{1 + \mathsf{fma}\left(\color{blue}{i \cdot i}, 0.08333333333333333, -0.5 \cdot i\right)} \cdot 100 \]
      *-commutative [=>] 99.9	\[ \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, \color{blue}{i \cdot -0.5}\right)} \cdot 100 \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 4 \cdot 10^{-183}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, i \cdot -0.5\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.0%
Cost	21768

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-183}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, i \cdot -0.5\right)}\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	21768

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-183}:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, i \cdot -0.5\right)}\\ \end{array} \]

Alternative 3
Accuracy	82.3%
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq 3.4 \cdot 10^{-185}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{+242}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{+293}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4
Accuracy	82.5%
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq 8 \cdot 10^{-186}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+257}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \mathbf{elif}\;i \leq 3.85 \cdot 10^{+292}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 5
Accuracy	84.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -1.9 \cdot 10^{-190} \lor \neg \left(i \leq 3.2 \cdot 10^{-186}\right):\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 6
Accuracy	82.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 2.65 \cdot 10^{-182}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \end{array} \]

Alternative 7
Accuracy	82.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 4.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \end{array} \]

Alternative 8
Accuracy	75.7%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;i \leq -3.05 \cdot 10^{-189} \lor \neg \left(i \leq 2.1 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 9
Accuracy	70.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-239} \lor \neg \left(n \leq 3 \cdot 10^{-207}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 10
Accuracy	70.0%
Cost	840

\[\begin{array}{l} t_0 := 1 + i \cdot -0.5\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-237}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t_0}\\ \end{array} \]

Alternative 11
Accuracy	64.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+28} \lor \neg \left(i \leq 1.42 \cdot 10^{-52}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 12
Accuracy	65.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \lor \neg \left(i \leq 8.8 \cdot 10^{-5}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13
Accuracy	68.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4 \lor \neg \left(i \leq 8.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14
Accuracy	56.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-242} \lor \neg \left(n \leq 4.3 \cdot 10^{-233}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 15
Accuracy	56.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-242} \lor \neg \left(n \leq 4.3 \cdot 10^{-233}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]

Alternative 16
Accuracy	3.0%
Cost	192

\[i \cdot -50 \]

Alternative 17
Accuracy	55.6%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023147 
(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))))