Compound Interest

Average Error: 47.2 → 11.8

Time: 18.9s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 530:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;i \leq 4.7 \cdot 10^{+225}:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot n}{i} \cdot \left(\log i - \log n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{100}}\\ \end{array} \]

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 530.0)
   (/ n (* (/ i (expm1 i)) 0.01))
   (if (<= i 4.7e+225)
     (* 100.0 (* (/ (* n n) i) (- (log i) (log n))))
     (/ (* n (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (/ i 100.0)))))

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 tmp;
	if (i <= 530.0) {
		tmp = n / ((i / expm1(i)) * 0.01);
	} else if (i <= 4.7e+225) {
		tmp = 100.0 * (((n * n) / i) * (log(i) - log(n)));
	} else {
		tmp = (n * (pow((1.0 + (i / n)), n) + -1.0)) / (i / 100.0);
	}
	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 <= 530.0) {
		tmp = n / ((i / Math.expm1(i)) * 0.01);
	} else if (i <= 4.7e+225) {
		tmp = 100.0 * (((n * n) / i) * (Math.log(i) - Math.log(n)));
	} else {
		tmp = (n * (Math.pow((1.0 + (i / n)), n) + -1.0)) / (i / 100.0);
	}
	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 <= 530.0:
		tmp = n / ((i / math.expm1(i)) * 0.01)
	elif i <= 4.7e+225:
		tmp = 100.0 * (((n * n) / i) * (math.log(i) - math.log(n)))
	else:
		tmp = (n * (math.pow((1.0 + (i / n)), n) + -1.0)) / (i / 100.0)
	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)
	tmp = 0.0
	if (i <= 530.0)
		tmp = Float64(n / Float64(Float64(i / expm1(i)) * 0.01));
	elseif (i <= 4.7e+225)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * n) / i) * Float64(log(i) - log(n))));
	else
		tmp = Float64(Float64(n * Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)) / Float64(i / 100.0));
	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_] := If[LessEqual[i, 530.0], N[(n / N[(N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] * 0.01), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.7e+225], N[(100.0 * N[(N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	47.2
Target	47.3
Herbie	11.8

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

Derivation

1. Split input into 3 regimes

2. if i < 530

   1. Initial program 49.8

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

   2. Simplified 49.7

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)} \]
      Proof
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) -1) i))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (neg.f64 1))) i))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n 1) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 (Rewrite=> *-rgt-identity_binary64 n) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))): 7 points increase in error, 1 points decrease in error
      (*.f64 100 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 0 points increase in error, 8 points decrease in error

   3. Taylor expanded in n around inf 45.0

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

   4. Simplified 10.1

      \[\leadsto 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]
      Proof
      (*.f64 100 (*.f64 n (/.f64 (expm1.f64 i) i))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 i) 1)) i))): 0 points increase in error, 2 points decrease in error

   5. Applied egg-rr 10.2

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

   6. Applied egg-rr 10.2

      \[\leadsto \frac{n}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}} \]

   if 530 < i < 4.70000000000000004e225

   1. Initial program 32.5

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

   2. Simplified 32.5

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)} \]
      Proof
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) -1) i))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (neg.f64 1))) i))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n 1) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 (Rewrite=> *-rgt-identity_binary64 n) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))): 7 points increase in error, 1 points decrease in error
      (*.f64 100 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 0 points increase in error, 8 points decrease in error

   3. Taylor expanded in n around 0 17.1

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

   4. Simplified 17.3

      \[\leadsto \color{blue}{100 \cdot \left(\frac{n \cdot n}{i} \cdot \left(\log i - \log n\right)\right)} \]
      Proof
      (*.f64 100 (*.f64 n (/.f64 (expm1.f64 i) i))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 i) 1)) i))): 0 points increase in error, 2 points decrease in error

   if 4.70000000000000004e225 < i

   1. Initial program 29.9

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

   2. Simplified 29.9

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)} \]
      Proof
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) -1) i))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (neg.f64 1))) i))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (*.f64 n (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n 1) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 8 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 (Rewrite=> *-rgt-identity_binary64 n) (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i))): 7 points increase in error, 1 points decrease in error
      (*.f64 100 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 0 points increase in error, 8 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 0 points increase in error, 8 points decrease in error

   3. Applied egg-rr 29.8

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 530:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;i \leq 4.7 \cdot 10^{+225}:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot n}{i} \cdot \left(\log i - \log n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{100}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.8
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;t_1 \leq 0.005:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 2
Error	11.8
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;t_1 \leq 0.005:\\ \;\;\;\;\frac{100 \cdot t_0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 3
Error	11.8
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;t_1 \leq 0.005:\\ \;\;\;\;\frac{n \cdot t_0}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 4
Error	11.9
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.004:\\ \;\;\;\;\frac{n}{\frac{1 + i \cdot \left(i \cdot 0.08333333333333333\right)}{100}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.0
Cost	7244

\[\begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-187}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.004:\\ \;\;\;\;\frac{n}{\frac{1 + i \cdot \left(i \cdot 0.08333333333333333\right)}{100}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \end{array} \]

Alternative 6
Error	12.0
Cost	7244

\[\begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot 0.01}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.004:\\ \;\;\;\;\frac{n}{\frac{1 + i \cdot \left(i \cdot 0.08333333333333333\right)}{100}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \end{array} \]

Alternative 7
Error	19.2
Cost	1480

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-192}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{1 + \left(\left(1 + i \cdot \left(i \cdot 0.08333333333333333\right)\right) + \left(-1 + i \cdot -0.5\right)\right)}{100}}\\ \end{array} \]

Alternative 8
Error	19.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-181}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 10^{-195}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 9
Error	21.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -3.45 \lor \neg \left(i \leq 152\right):\\ \;\;\;\;1200 \cdot \frac{n}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 10
Error	21.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.95 \lor \neg \left(i \leq 240\right):\\ \;\;\;\;1200 \cdot \frac{n}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 11
Error	20.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq 1.6 \lor \neg \left(i \leq 9.8 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;1200 \cdot \frac{n}{i \cdot i}\\ \end{array} \]

Alternative 12
Error	19.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -2.55 \cdot 10^{-195} \lor \neg \left(n \leq 6.5 \cdot 10^{-196}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 13
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 14
Error	28.8
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2022343 
(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))))