Compound Interest

Average Error: 48.2 → 1.0

Time: 15.4s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-241}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;\frac{t_0 \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

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)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 1e-241)
     (* n (/ (expm1 (* n (log1p (/ i n)))) (/ i 100.0)))
     (if (<= t_1 20000000000000.0)
       (/ (* t_0 100.0) (/ i n))
       (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -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;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-241) {
		tmp = n * (expm1((n * log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= 20000000000000.0) {
		tmp = (t_0 * 100.0) / (i / n);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	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 t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-241) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= 20000000000000.0) {
		tmp = (t_0 * 100.0) / (i / n);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	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):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= 1e-241:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) / (i / 100.0))
	elif t_1 <= 20000000000000.0:
		tmp = (t_0 * 100.0) / (i / n)
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -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(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-241)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / 100.0)));
	elseif (t_1 <= 20000000000000.0)
		tmp = Float64(Float64(t_0 * 100.0) / Float64(i / n));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -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[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-241], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20000000000000.0], N[(N[(t$95$0 * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	48.2
Target	47.8
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.9999999999999997e-242

   1. Initial program 46.6

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

   2. Applied egg-rr 2.2

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right)} \]

   3. Simplified 1.1

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}} \]
      Proof
      (*.f64 n (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (/.f64 i 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 n (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i) 100))): 16 points increase in error, 17 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i)) 100)): 14 points increase in error, 19 points decrease in error
      (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 n (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n))))) i)) 100): 36 points increase in error, 11 points decrease in error
      (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 n i) (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))))) 100): 60 points increase in error, 33 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (/.f64 n i))) 100): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (*.f64 (/.f64 n i) 100))): 22 points increase in error, 13 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (*.f64 (/.f64 n i) 100)))): 0 points increase in error, 0 points decrease in error

   if 9.9999999999999997e-242 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e13

   1. Initial program 2.5

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

   2. Simplified 2.4

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{n}}} \]
      Proof
      (/.f64 (*.f64 100 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) -1)) (/.f64 i n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 100 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 i n)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 100 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))) (/.f64 i n)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 7 points increase in error, 3 points decrease in error

   if 2e13 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.7

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

   2. Applied egg-rr 62.7

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

   3. Applied egg-rr 63.7

      \[\leadsto \color{blue}{0 + 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}} \]

   4. Simplified 62.7

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(100 \cdot n\right)} \]
      Proof
      (*.f64 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i) (*.f64 100 n)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i) (Rewrite=> *-commutative_binary64 (*.f64 n 100))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i) n) 100)): 15 points increase in error, 21 points decrease in error
      (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (/.f64 i n))) 100): 52 points increase in error, 19 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 100 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (/.f64 i n)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 100 (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) (/.f64 i n))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 62.7

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

   6. Taylor expanded in i around 0 0.3

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

   7. Simplified 0.3

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
      Proof
      (+.f64 1 (*.f64 i (+.f64 (/.f64 1/2 n) -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 i (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) n) -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 i (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 n))) -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 i (+.f64 (*.f64 1/2 (/.f64 1 n)) (Rewrite<= metadata-eval (neg.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 i (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 1/2 (/.f64 1 n)) 1/2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-241}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 20000000000000:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-241}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;\frac{t_0 \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 2
Error	9.0
Cost	7500

\[\begin{array}{l} t_0 := \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{if}\;i \leq -0.036:\\ \;\;\;\;\frac{100 \cdot \left(-1 + e^{i}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 130:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\log \left(\frac{i}{n}\right) \cdot \frac{n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.0
Cost	7500

\[\begin{array}{l} t_0 := \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{if}\;i \leq -0.048:\\ \;\;\;\;\frac{100 \cdot \left(-1 + e^{i}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 150:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+47}:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot n}{i} \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	8.2
Cost	7112

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -255:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	20.5
Cost	972

\[\begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{+48}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 3.9 \cdot 10^{-227}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{i}{i \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	19.4
Cost	840

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{-212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.0
Cost	832

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

Alternative 8
Error	22.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-16}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 9
Error	21.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+41}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -5.9 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 10
Error	23.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-210}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11
Error	51.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022330 
(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))))