Compound Interest

Average Error: 47.7 → 0.6

Time: 25.5s

Precision: binary64

Cost: 22024

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 10000:\\ \;\;\;\;\frac{n}{i} \cdot \left(t_0 \cdot 100\right) + \frac{n}{i} \cdot -100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

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)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 10000.0)
       (+ (* (/ n i) (* t_0 100.0)) (* (/ n i) -100.0))
       (/ 100.0 (+ (/ 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 = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= 10000.0) {
		tmp = ((n / i) * (t_0 * 100.0)) + ((n / i) * -100.0);
	} else {
		tmp = 100.0 / ((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 = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= 10000.0) {
		tmp = ((n / i) * (t_0 * 100.0)) + ((n / i) * -100.0);
	} else {
		tmp = 100.0 / ((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 = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= 10000.0:
		tmp = ((n / i) * (t_0 * 100.0)) + ((n / i) * -100.0)
	else:
		tmp = 100.0 / ((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(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= 10000.0)
		tmp = Float64(Float64(Float64(n / i) * Float64(t_0 * 100.0)) + Float64(Float64(n / i) * -100.0));
	else
		tmp = Float64(100.0 / 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[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 10000.0], N[(N[(N[(n / i), $MachinePrecision] * N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] + N[(N[(n / i), $MachinePrecision] * -100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.7
Target	47.7
Herbie	0.6

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

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 46.1

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

   2. Applied egg-rr 0.2

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

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e4

   1. Initial program 3.3

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

   2. Simplified 3.3

      \[\leadsto \color{blue}{\frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)} \]
      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)))): 2 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, 2 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, 57 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))) i)): 6 points increase in error, 5 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 n (/.f64 (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 3 points increase in error, 4 points decrease in error
      (*.f64 n (Rewrite<= associate-*r/_binary64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 8 points increase in error, 3 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))): 2 points increase in error, 7 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 55 points increase in error, 3 points decrease in error

   3. Applied egg-rr 3.3

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

   4. Applied egg-rr 3.4

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

   if 1e4 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.0

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

   2. Taylor expanded in n around inf 62.0

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

   3. Applied egg-rr 13.9

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

   4. Taylor expanded in i around 0 1.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10000:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100\right) + \frac{n}{i} \cdot -100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	22024

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 10000:\\ \;\;\;\;\frac{n}{i} \cdot \left(t_0 \cdot 100\right) + \frac{n}{i} \cdot -100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternative 2
Error	11.7
Cost	7816

\[\begin{array}{l} \mathbf{if}\;i \leq -9.8 \cdot 10^{-62}:\\ \;\;\;\;100 \cdot \frac{\left(1 + n \cdot \mathsf{expm1}\left(i\right)\right) + -1}{i}\\ \mathbf{elif}\;i \leq 880000000:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 + \frac{1}{n} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100\right) + \frac{n}{i} \cdot -100\\ \end{array} \]

Alternative 3
Error	11.7
Cost	7560

\[\begin{array}{l} \mathbf{if}\;i \leq -9.8 \cdot 10^{-62}:\\ \;\;\;\;100 \cdot \frac{\left(1 + n \cdot \mathsf{expm1}\left(i\right)\right) + -1}{i}\\ \mathbf{elif}\;i \leq 880000000:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 + \frac{1}{n} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100\right)\\ \end{array} \]

Alternative 4
Error	12.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{100}{\frac{i}{\left(1 + n \cdot \mathsf{expm1}\left(i\right)\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternative 5
Error	12.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \frac{\left(1 + n \cdot \mathsf{expm1}\left(i\right)\right) + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	7112

\[\begin{array}{l} t_0 := n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \mathbf{if}\;n \leq -1.42 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-147}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.3
Cost	972

\[\begin{array}{l} t_0 := 0 \cdot \frac{n}{i}\\ \mathbf{if}\;i \leq -6.8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -2.370876729336427 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+40}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50 + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	19.3
Cost	968

\[\begin{array}{l} t_0 := \frac{100}{\frac{1}{n} + \frac{i}{n} \cdot -0.5}\\ \mathbf{if}\;n \leq -1.42 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.86 \cdot 10^{-221}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.4
Cost	844

\[\begin{array}{l} t_0 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -2.370876729336427 \cdot 10^{-213}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	22.4
Cost	844

\[\begin{array}{l} t_0 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -2.370876729336427 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -2.370876729336427 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \]

Alternative 12
Error	20.3
Cost	844

\[\begin{array}{l} t_0 := 0 \cdot \frac{n}{i}\\ \mathbf{if}\;i \leq -6.8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -2.370876729336427 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+40}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	27.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-217}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-294}:\\ \;\;\;\;n \cdot \left(16.666666666666668 \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 14
Error	27.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-265}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-294}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 15
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 16
Error	28.1
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2022308 
(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))))