Compound Interest

Average Accuracy: 25.0% → 98.6%

Time: 21.3s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 10^{-14}:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

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 1e-14)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (* 100.0 (/ n (+ 1.0 (* i -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 <= 1e-14) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -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 <= 1e-14) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -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 <= 1e-14:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 * (n / (1.0 + (i * -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 <= 1e-14)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -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, 1e-14], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	25.0%
Target	26.7%
Herbie	98.6%

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

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 27.3%

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

   2. Applied egg-rr 99.6%

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

      [Start] 27.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      *-commutative [=>] 27.3	\[ \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]
      associate-*l/ [=>] 27.3	\[ \color{blue}{\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}} \]
      pow-to-exp [=>] 27.3	\[ \frac{\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{\frac{i}{n}} \]
      expm1-def [=>] 39.9	\[ \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot 100}{\frac{i}{n}} \]
      *-commutative [=>] 39.9	\[ \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log1p-udef [<=] 99.6	\[ \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.99999999999999999e-15

   1. Initial program 94.5%

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

   2. Simplified 94.6%

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

      [Start] 94.5	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 94.6	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 94.6	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      distribute-lft-in [=>] 94.4	\[ \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]
      fma-def [=>] 94.6	\[ \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 94.6	\[ \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]
      metadata-eval [=>] 94.6	\[ \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Applied egg-rr 94.4%

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

      [Start] 94.6	\[ \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}} \]
      fma-udef [=>] 94.4	\[ \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

   if 9.99999999999999999e-15 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 3.6%

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

   2. Taylor expanded in n around inf 3.6%

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

   3. Simplified 78.5%

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

      [Start] 3.6	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      *-commutative [=>] 3.6	\[ \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]
      associate-/l* [=>] 3.6	\[ \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]
      expm1-def [=>] 78.5	\[ \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Taylor expanded in i around 0 96.2%

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

   5. Simplified 96.2%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]
      Proof

      [Start] 96.2	\[ \frac{n}{1 + -0.5 \cdot i} \cdot 100 \]
      *-commutative [=>] 96.2	\[ \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-14}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq 10^{-14}:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 10^{-14}:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3
Accuracy	97.8%
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq 10^{-14}:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4
Accuracy	83.1%
Cost	7376

\[\begin{array}{l} t_0 := n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := \frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-224}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	83.1%
Cost	7376

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ t_1 := \frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-60}:\\ \;\;\;\;n \cdot \frac{100}{t_0}\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	80.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{-12}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.08:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(\frac{i}{n} \cdot \left(-0.5 + \frac{0.3333333333333333}{n}\right) + \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	68.6%
Cost	1868

\[\begin{array}{l} \mathbf{if}\;i \leq -0.029:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -5.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;i \leq 0.08:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(\frac{i}{n} \cdot \left(-0.5 + \frac{0.3333333333333333}{n}\right) + \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	68.5%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;i \leq -0.04:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-139}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;i \leq 0.185:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	68.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;i \leq -0.038:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -2.46 \cdot 10^{-138}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;i \leq 0.07:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	67.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0045:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.112:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + \frac{-50}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	67.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0068:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.029:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	68.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -0.044:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.055:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	67.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -0.039:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.13:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	19.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023151 
(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))))