Compound Interest

Average Error: 48.2 → 0.5

Time: 16.4s

Precision: binary64

Cost: 21896

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 100000000:\\ \;\;\;\;n \cdot \left(\frac{t_0}{i \cdot 0.01} + \frac{-100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

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 0.0)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 100000000.0)
       (* n (+ (/ t_0 (* i 0.01)) (/ -100.0 i)))
       (* 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 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 100000000.0) {
		tmp = n * ((t_0 / (i * 0.01)) + (-100.0 / i));
	} else {
		tmp = 100.0 * ((1.0 + n) + -1.0);
	}
	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 t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 100000000.0) {
		tmp = n * ((t_0 / (i * 0.01)) + (-100.0 / i));
	} else {
		tmp = 100.0 * ((1.0 + n) + -1.0);
	}
	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):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= 100000000.0:
		tmp = n * ((t_0 / (i * 0.01)) + (-100.0 / i))
	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 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= 100000000.0)
		tmp = Float64(n * Float64(Float64(t_0 / Float64(i * 0.01)) + Float64(-100.0 / i)));
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 + 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, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 100000000.0], N[(n * N[(N[(t$95$0 / N[(i * 0.01), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(1.0 + n), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	48.2
Target	47.6
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 46.6

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

   2. Applied egg-rr 0.3

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e8

   1. Initial program 3.4

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

   2. Applied egg-rr 53.9

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

   3. Simplified 53.9

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}} \]
      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))): 20 points increase in error, 18 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 n (/.f64 (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))) i)) 100)): 10 points increase in error, 13 points decrease in error
      (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 n (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n))))) i)) 100): 31 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 n i) (expm1.f64 (*.f64 n (log1p.f64 (/.f64 i n)))))) 100): 52 points increase in error, 27 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))): 16 points increase in error, 9 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

   4. Applied egg-rr 3.3

      \[\leadsto n \cdot \color{blue}{\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot 0.01} - \frac{100}{i}\right)} \]

   if 1e8 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.5

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

   2. Taylor expanded in i around 0 49.8

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

   3. Applied egg-rr 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 100000000:\\ \;\;\;\;n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot 0.01} + \frac{-100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 100000000:\\ \;\;\;\;n \cdot \left(\frac{t_0}{i \cdot 0.01} + \frac{-100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq 100000000:\\ \;\;\;\;n \cdot \left(\frac{t_0}{i \cdot 0.01} + \frac{-100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternative 3
Error	1.0
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{0.01}}}\\ \mathbf{elif}\;t_1 \leq 100000000:\\ \;\;\;\;n \cdot \left(\frac{t_0}{i \cdot 0.01} + \frac{-100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternative 4
Error	11.6
Cost	7560

\[\begin{array}{l} \mathbf{if}\;i \leq 20:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\left(1 + \frac{n}{i}\right) + -1\right) - \frac{n}{i}\right)\\ \end{array} \]

Alternative 5
Error	12.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 20:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\left(1 + \frac{n}{i}\right) + -1\right) - \frac{n}{i}\right)\\ \end{array} \]

Alternative 6
Error	17.6
Cost	1228

\[\begin{array}{l} t_0 := 100 \cdot \left(\left(\left(1 + \frac{n}{i}\right) + -1\right) - \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -3.25 \cdot 10^{+240}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-62}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \mathbf{elif}\;i \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.5
Cost	1032

\[\begin{array}{l} \mathbf{if}\;i \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \mathbf{elif}\;i \leq 7.4:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{-1}{-i} - \frac{n}{i}\right)\\ \end{array} \]

Alternative 8
Error	19.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{-62}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \mathbf{elif}\;i \leq 10.8:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	19.8
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \left(\left(1 + n\right) + -1\right)\\ \mathbf{if}\;i \leq -8.2 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-169}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	19.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \mathbf{elif}\;i \leq 6:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	20.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -2.25 \cdot 10^{+24}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 4:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	50.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(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))))