Compound Interest

Average Error: 47.8 → 0.7

Time: 25.7s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\frac{n}{i} \cdot \left(t_0 \cdot 100 + -100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

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 2.0)
       (* (/ n i) (+ (* t_0 100.0) -100.0))
       (* 100.0 (/ 1.0 (+ (/ 1.0 n) (* (/ i 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);
	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 <= 2.0) {
		tmp = (n / i) * ((t_0 * 100.0) + -100.0);
	} else {
		tmp = 100.0 * (1.0 / ((1.0 / n) + ((i / 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);
	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 <= 2.0) {
		tmp = (n / i) * ((t_0 * 100.0) + -100.0);
	} else {
		tmp = 100.0 * (1.0 / ((1.0 / n) + ((i / 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)
	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 <= 2.0:
		tmp = (n / i) * ((t_0 * 100.0) + -100.0)
	else:
		tmp = 100.0 * (1.0 / ((1.0 / n) + ((i / 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(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 <= 2.0)
		tmp = Float64(Float64(n / i) * Float64(Float64(t_0 * 100.0) + -100.0));
	else
		tmp = Float64(100.0 * Float64(1.0 / Float64(Float64(1.0 / n) + Float64(Float64(i / 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[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, 2.0], N[(N[(n / i), $MachinePrecision] * N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(1.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.8
Target	46.9
Herbie	0.7

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

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 46.4

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

   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)) < 2

   1. Initial program 3.8

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

   2. Simplified 3.8

      \[\leadsto \color{blue}{\frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)} \]
      Proof
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) -100)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (*.f64 -1 100)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n)) (*.f64 (neg.f64 1) 100)))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 n i) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 100)) (*.f64 (neg.f64 1) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= distribute-rgt-in_binary64 (*.f64 100 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (neg.f64 1))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (*.f64 100 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 i (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))))): 4 points increase in error, 52 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))) i)): 8 points increase in error, 4 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 n (/.f64 (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 4 points increase in error, 8 points decrease in error
      (*.f64 n (Rewrite<= associate-*r/_binary64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 6 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)) n)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 100 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 4 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)))): 47 points increase in error, 5 points decrease in error

   3. Applied egg-rr 3.8

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

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

   1. Initial program 62.2

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

   2. Taylor expanded in n around inf 62.2

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

   3. Simplified 62.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
      Proof
      (expm1.f64 i): 0 points increase in error, 0 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 i) 1)): 77 points increase in error, 89 points decrease in error

   4. Applied egg-rr 13.1

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{1}{n}}\right)} \]

   5. Applied egg-rr 12.9

      \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{1}{n}}{\frac{\mathsf{expm1}\left(i\right)}{i}}}} \]

   6. Taylor expanded in i around 0 1.7

      \[\leadsto 100 \cdot \frac{1}{\color{blue}{\frac{1}{n} + -0.5 \cdot \frac{i}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.5
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\frac{1}{n}}{\frac{\mathsf{expm1}\left(i\right)}{i}}}\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\frac{n}{i} \cdot \left(t_0 \cdot 100 + -100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternative 2
Error	2.0
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq 2:\\ \;\;\;\;\frac{n}{i} \cdot \left(t_0 \cdot 100 + -100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternative 3
Error	8.7
Cost	13900

\[\begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)}{i}\\ \mathbf{if}\;i \leq -1.12 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.9 \cdot 10^{-71}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100\right)\\ \end{array} \]

Alternative 4
Error	8.5
Cost	13900

\[\begin{array}{l} t_0 := \frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{if}\;i \leq -1.12 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.9 \cdot 10^{-71}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100\right)\\ \end{array} \]

Alternative 5
Error	10.6
Cost	7376

\[\begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ t_1 := 100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{if}\;n \leq -4 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.25 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-210}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.9
Cost	7112

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -8.8 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 0.0023:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 8
Error	19.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 9
Error	19.1
Cost	1096

\[\begin{array}{l} t_0 := 100 \cdot \frac{1}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-210}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;i \leq -4.55 \cdot 10^{+30}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	19.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	19.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+33}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0009:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	51.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022316 
(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))))