Compound Interest

Average Error: 74.88% → 17.3%

Time: 19.3s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot t_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

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

Target

Original	74.88%
Target	74.23%
Herbie	17.3%

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

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 72.4

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

   2. Simplified 72.66

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

      [Start] 72.4	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 72.66	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 72.66	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 72.66	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 72.66	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 72.66	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 72.66	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 72.66	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 66.2

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

   4. Simplified 23.06

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

      [Start] 66.2	\[ 100 \cdot \left(n \cdot \frac{e^{i} - 1}{i}\right) \]
      expm1-def [=>] 23.06	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e13

   1. Initial program 4.16

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

   2. Simplified 4.19

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

      [Start] 4.16	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 4.19	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 4.19	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 4.19	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 4.19	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 4.19	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 4.19	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 4.19	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Applied egg-rr 4.09

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

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

   1. Initial program 99.46

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

   2. Simplified 97.91

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

      [Start] 99.46	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 97.91	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 97.91	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 97.91	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 97.91	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 97.91	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 97.91	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 97.91	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 97.87

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

   4. Simplified 20.72

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

      [Start] 97.87	\[ 100 \cdot \left(n \cdot \frac{e^{i} - 1}{i}\right) \]
      expm1-def [=>] 20.72	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]

   5. Applied egg-rr 20.87

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

   6. Taylor expanded in i around 0 20.11

      \[\leadsto \frac{n}{\color{blue}{0.01}} \]

   7. Applied egg-rr 0.62

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 20000000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.3%
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 2
Error	17.3%
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;n \cdot \left(t_0 \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 3
Error	17.55%
Cost	7376

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ t_1 := \frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.9 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-131}:\\ \;\;\;\;n \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 0.02:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	30.95%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;n \leq -5.7 \cdot 10^{-238}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-131}:\\ \;\;\;\;n \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{\left(i \cdot -0.005\right) \cdot \left(i \cdot -0.005\right) + -0.0001}{i \cdot -0.005 + -0.01}}\\ \end{array} \]

Alternative 5
Error	31.22%
Cost	836

\[\begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-131}:\\ \;\;\;\;n \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \]

Alternative 6
Error	32.22%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.46 \cdot 10^{-16} \lor \neg \left(i \leq 1.35 \cdot 10^{+21}\right):\\ \;\;\;\;n \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 7
Error	30.48%
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-238} \lor \neg \left(n \leq 8.2 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{0}{i}\\ \end{array} \]

Alternative 8
Error	32.37%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+27}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \end{array} \]

Alternative 9
Error	31.18%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{0}{i}\\ \end{array} \]

Alternative 10
Error	32.95%
Cost	585

\[\begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.75 \cdot 10^{+24}\right):\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11
Error	32.75%
Cost	584

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \end{array} \]

Alternative 12
Error	97.02%
Cost	192

\[i \cdot -50 \]

Alternative 13
Error	43.66%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023090 
(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))))