Compound Interest

Average Error: 47.4 → 0.4

Time: 21.4s

Precision: binary64

Cost: 21896

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\frac{1}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

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)
     (* 100.0 (/ (/ 1.0 (/ 1.0 (expm1 (* n (log1p (/ i n)))))) (/ i n)))
     (if (<= t_1 2e+15)
       (* 100.0 (- (/ t_0 (/ i n)) (/ n i)))
       (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334))))))))

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 * ((1.0 / (1.0 / expm1((n * log1p((i / n)))))) / (i / n));
	} else if (t_1 <= 2e+15) {
		tmp = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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(Float64(1.0 / Float64(1.0 / expm1(Float64(n * log1p(Float64(i / n)))))) / Float64(i / n)));
	elseif (t_1 <= 2e+15)
		tmp = Float64(100.0 * Float64(Float64(t_0 / Float64(i / n)) - Float64(n / i)));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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[(1.0 / N[(1.0 / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+15], N[(100.0 * N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.4
Target	47.1
Herbie	0.4

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

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 46.0

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

   2. Applied egg-rr 0.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}}{\frac{i}{n}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e15

   1. Initial program 1.6

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

   2. Applied egg-rr 53.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}}{\frac{i}{n}} \]

   3. Applied egg-rr 1.5

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

   if 2e15 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.7

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

   2. Simplified 63.7

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

      [Start] 63.7	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 63.7	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 63.7	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 63.7	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in n around inf 63.8

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

   4. Simplified 63.8

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

      [Start] 63.8	\[ \frac{100 \cdot \left(e^{i} - 1\right)}{\frac{i}{n}} \]
      expm1-def [=>] 63.8	\[ \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   5. Applied egg-rr 13.8

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

   6. Applied egg-rr 13.9

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

   7. Taylor expanded in i around 0 0.4

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

   8. Simplified 0.4

      \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
      Proof

      [Start] 0.4	\[ \frac{n}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)} \]
      fma-def [=>] 0.4	\[ \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
      *-commutative [=>] 0.4	\[ \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]
      unpow2 [=>] 0.4	\[ \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\frac{1}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 4
Error	10.9
Cost	7632

\[\begin{array}{l} t_0 := \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}\\ \mathbf{elif}\;n \leq -7 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-206}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)}{i}\\ \end{array} \]

Alternative 5
Error	12.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{-206}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;i \leq 560:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	12.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)}{i}\\ \mathbf{elif}\;i \leq 510:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	12.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -8.2 \cdot 10^{-57}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-207}:\\ \;\;\;\;\frac{i \cdot n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 24:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + i \cdot 0.16666666666666666\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	12.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -1.65 \cdot 10^{-206}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;i \leq 41:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + i \cdot 0.16666666666666666\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	20.5
Cost	1228

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-207}:\\ \;\;\;\;\frac{i \cdot n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 70:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + i \cdot 0.16666666666666666\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	20.5
Cost	972

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{i \cdot n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 35:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	20.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{i \cdot n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	19.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-229} \lor \neg \left(n \leq 1.7 \cdot 10^{-148}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -0.94:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 6.4:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	21.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.42:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Error	51.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023034 
(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))))