Compound Interest

Average Error: 47.9 → 0.4

Time: 21.9s

Precision: binary64

Cost: 28168

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 1000000000000:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, t_0, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

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 (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 4e-259)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 1000000000000.0)
       (/ 1.0 (/ (/ i n) (fma 100.0 t_0 -100.0)))
       (* 100.0 (+ 1.0 (+ n -1.0)))))))

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);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 4e-259) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= 1000000000000.0) {
		tmp = 1.0 / ((i / n) / fma(100.0, t_0, -100.0));
	} else {
		tmp = 100.0 * (1.0 + (n + -1.0));
	}
	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(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 4e-259)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= 1000000000000.0)
		tmp = Float64(1.0 / Float64(Float64(i / n) / fma(100.0, t_0, -100.0)));
	else
		tmp = Float64(100.0 * Float64(1.0 + Float64(n + -1.0)));
	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[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-259], 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, 1000000000000.0], N[(1.0 / N[(N[(i / n), $MachinePrecision] / N[(100.0 * t$95$0 + -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(1.0 + N[(n + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.9
Target	47.5
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 4.0000000000000003e-259

   1. Initial program 46.1

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 0.3

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

   if 4.0000000000000003e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e12

   1. Initial program 2.9

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

   2. Simplified 2.7

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

      [Start] 2.9	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 2.8	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      associate-*r* [=>] 2.8	\[ \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]
      *-lft-identity [<=] 2.8	\[ \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot \color{blue}{\left(1 \cdot n\right)} \]
      associate-*r* [=>] 2.8	\[ \color{blue}{\left(\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot 1\right) \cdot n} \]
      *-commutative [=>] 2.8	\[ \color{blue}{n \cdot \left(\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot 1\right)} \]
      *-commutative [=>] 2.8	\[ n \cdot \color{blue}{\left(1 \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 2.8	\[ n \cdot \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      associate-*r/ [=>] 2.7	\[ n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]
      sub-neg [=>] 2.7	\[ n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]
      distribute-lft-in [=>] 2.7	\[ n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]
      fma-def [=>] 2.7	\[ n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]
      metadata-eval [=>] 2.7	\[ n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]
      metadata-eval [=>] 2.7	\[ n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Applied egg-rr 2.8

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

   if 1e12 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.8

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

   2. Taylor expanded in i around 0 49.2

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 4 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 1000000000000:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	28040

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 1000000000000:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 1000000000000:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 1000000000000:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

Alternative 4
Error	13.0
Cost	7684

\[\begin{array}{l} \mathbf{if}\;i \leq 370:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100\right) \cdot \frac{n}{i} + -100 \cdot \frac{n}{i}\\ \end{array} \]

Alternative 5
Error	13.0
Cost	7428

\[\begin{array}{l} \mathbf{if}\;i \leq 550:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \end{array} \]

Alternative 6
Error	13.0
Cost	7428

\[\begin{array}{l} \mathbf{if}\;i \leq 520:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \end{array} \]

Alternative 7
Error	13.0
Cost	7428

\[\begin{array}{l} \mathbf{if}\;i \leq 520:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \end{array} \]

Alternative 8
Error	11.6
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{-263}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 9.6 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{0.5 + i \cdot -0.25}}{i + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	11.6
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -2.1 \cdot 10^{-263}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{0.5 + i \cdot -0.25}}{i + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-266} \lor \neg \left(n \leq 2.2 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{\frac{n \cdot 100}{0.5 + i \cdot -0.25}}{i + 2}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \end{array} \]

Alternative 11
Error	19.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;i \leq -0.36:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;i \leq 1.75 \cdot 10^{-121}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(100 \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(-1 + \left(1 + n\right)\right)\\ \end{array} \]

Alternative 12
Error	21.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;i \leq 1.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{n}{i + 2} \cdot \frac{100}{\frac{1}{i + 2}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(-1 + \left(1 + n\right)\right)\\ \end{array} \]

Alternative 13
Error	19.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;i \leq -1.9 \cdot 10^{-179}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{-122}:\\ \;\;\;\;n \cdot \frac{200}{i + 2}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(-1 + \left(1 + n\right)\right)\\ \end{array} \]

Alternative 14
Error	20.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -9.5 \cdot 10^{+53} \lor \neg \left(i \leq 1.75 \cdot 10^{-121}\right):\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 15
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -9.5 \cdot 10^{+53}:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;i \leq 1.75 \cdot 10^{-121}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(-1 + \left(1 + n\right)\right)\\ \end{array} \]

Alternative 16
Error	19.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -0.47:\\ \;\;\;\;100 \cdot \left(1 + \left(n + -1\right)\right)\\ \mathbf{elif}\;i \leq 1.75 \cdot 10^{-121}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(-1 + \left(1 + n\right)\right)\\ \end{array} \]

Alternative 17
Error	23.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;i \leq -2.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{n}{i} \cdot 200\\ \mathbf{elif}\;i \leq 290:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{200}{\frac{i}{n}}\\ \end{array} \]

Alternative 18
Error	26.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{+140}:\\ \;\;\;\;\frac{n}{i} \cdot 200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 19
Error	28.4
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023066 
(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))))