Compound Interest

Average Error: 25.9% → 80.3%

Time: 23.6s

Precision: binary64

Cost: 7560.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -7 \cdot 10^{-140}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 12:\\ \;\;\;\;i \cdot \left(n \cdot 50 + -50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (i n)
 :precision binary64
 (if (<= i -7e-140)
   (/ 100.0 (/ i (* n (expm1 i))))
   (if (<= i 12.0)
     (+ (* i (+ (* n 50.0) -50.0)) (* 100.0 n))
     (* 100.0 (* n (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) 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 tmp;
	if (i <= -7e-140) {
		tmp = 100.0 / (i / (n * expm1(i)));
	} else if (i <= 12.0) {
		tmp = (i * ((n * 50.0) + -50.0)) + (100.0 * n);
	} else {
		tmp = 100.0 * (n * ((pow((1.0 + (i / n)), n) + -1.0) / i));
	}
	return tmp;
}

Java

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

↓

public static double code(double i, double n) {
	double tmp;
	if (i <= -7e-140) {
		tmp = 100.0 / (i / (n * Math.expm1(i)));
	} else if (i <= 12.0) {
		tmp = (i * ((n * 50.0) + -50.0)) + (100.0 * n);
	} else {
		tmp = 100.0 * (n * ((Math.pow((1.0 + (i / n)), n) + -1.0) / i));
	}
	return tmp;
}

Python

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

↓

def code(i, n):
	tmp = 0
	if i <= -7e-140:
		tmp = 100.0 / (i / (n * math.expm1(i)))
	elif i <= 12.0:
		tmp = (i * ((n * 50.0) + -50.0)) + (100.0 * n)
	else:
		tmp = 100.0 * (n * ((math.pow((1.0 + (i / n)), n) + -1.0) / 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)
	tmp = 0.0
	if (i <= -7e-140)
		tmp = Float64(100.0 / Float64(i / Float64(n * expm1(i))));
	elseif (i <= 12.0)
		tmp = Float64(Float64(i * Float64(Float64(n * 50.0) + -50.0)) + Float64(100.0 * n));
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / 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_] := If[LessEqual[i, -7e-140], N[(100.0 / N[(i / N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 12.0], N[(N[(i * N[(N[(n * 50.0), $MachinePrecision] + -50.0), $MachinePrecision]), $MachinePrecision] + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	25.9%
Target	25.5%
Herbie	80.3%

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

Derivation

1. Split input into 3 regimes

2. if i < -6.9999999999999996e-140

   1. Initial program 40.8

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

   2. Simplified 40.3

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

      [Start] 40.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 40.3	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 40.3	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 40.3	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 40.3	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 40.3	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 40.3	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 40.3	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 57.6

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

   4. Simplified 79.5

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

      [Start] 57.6	\[ 100 \cdot \left(n \cdot \frac{e^{i} - 1}{i}\right) \]
      expm1-def [=>] 79.5	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]

   5. Applied egg-rr 81.7

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

   if -6.9999999999999996e-140 < i < 12

   1. Initial program 8.3

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

   2. Simplified 8.8

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

      [Start] 8.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 8.8	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 8.8	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 8.8	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 8.8	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 8.8	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 8.8	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 8.8	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in i around 0 87.0

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

   4. Simplified 87.0

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

      [Start] 87.0	\[ 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n \]
      distribute-lft-out [=>] 87.0	\[ \color{blue}{100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n\right)} \]
      +-commutative [<=] 87.0	\[ 100 \cdot \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
      associate-*r/ [=>] 87.0	\[ 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]
      metadata-eval [=>] 87.0	\[ 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

   5. Taylor expanded in n around 0 87.0

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + 0.5 \cdot i\right)\right) + -50 \cdot i} \]

   6. Taylor expanded in i around 0 87.0

      \[\leadsto \color{blue}{i \cdot \left(50 \cdot n - 50\right) + 100 \cdot n} \]

   if 12 < i

   1. Initial program 51.5

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

   2. Simplified 51.5

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

      [Start] 51.5	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 51.5	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 51.5	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 51.5	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 51.5	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 51.5	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 51.5	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 51.5	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 80.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7 \cdot 10^{-140}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 12:\\ \;\;\;\;i \cdot \left(n \cdot 50 + -50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	80.2%
Cost	6980.00

\[\begin{array}{l} \mathbf{if}\;i \leq 520:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{100 \cdot n + 100 \cdot \left(\left(i \cdot n\right) \cdot \left(-0.5 - \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 2
Error	80.7%
Cost	6980.00

\[\begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 0.18:\\ \;\;\;\;i \cdot \left(n \cdot 50 + -50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{100 \cdot n + 100 \cdot \left(\left(i \cdot n\right) \cdot \left(-0.5 - \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 3
Error	69.6%
Cost	1872.00

\[\begin{array}{l} t_0 := \frac{n \cdot \left(n \cdot 10000\right)}{100 \cdot n + 100 \cdot \left(\left(i \cdot n\right) \cdot \left(-0.5 - \frac{-0.5}{n}\right)\right)}\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;n \leq -8 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-283}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 4
Error	68.2%
Cost	969.00

\[\begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{-5} \lor \neg \left(i \leq 23\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(-0.5 + n \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5
Error	68.2%
Cost	969.00

\[\begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{-5} \lor \neg \left(i \leq 3.1\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50 + -50\right) + 100 \cdot n\\ \end{array} \]

Alternative 6
Error	68.2%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{-5} \lor \neg \left(i \leq 28\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	64.0%
Cost	840.00

\[\begin{array}{l} \mathbf{if}\;i \leq -1.4:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 170:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array} \]

Alternative 8
Error	63.6%
Cost	713.00

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \lor \neg \left(i \leq 210\right):\\ \;\;\;\;i \cdot \left(n \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 9
Error	64.0%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 150:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array} \]

Alternative 10
Error	3.0%
Cost	192.00

\[i \cdot -50 \]

Alternative 11
Error	55.4%
Cost	192.00

\[100 \cdot n \]

Reproduce

herbie shell --seed 2023093 
(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))))