Compound Interest

Average Accuracy: 26.6% → 77.6%

Time: 26.3s

Precision: binary64

Cost: 26824

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{i}\right), \log \left(\frac{-1}{n}\right)\right)\right)}}\\ \mathbf{elif}\;i \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

↓

(FPCore (i n)
 :precision binary64
 (if (<= i -5.1e+33)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= i -3.1e-53)
     (*
      100.0
      (/ n (/ i (expm1 (* n (fma -1.0 (log (/ -1.0 i)) (log (/ -1.0 n))))))))
     (if (<= i 0.000155)
       (fma
        100.0
        (* n (* 0.3333333333333333 (* i (/ (/ i n) n))))
        (* 100.0 (+ n (* n (* i (- 0.5 (/ 0.5 n)))))))
       (* (/ n i) (* 100.0 (* n (- (log i) (log n)))))))))

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 tmp;
	if (i <= -5.1e+33) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (i <= -3.1e-53) {
		tmp = 100.0 * (n / (i / expm1((n * fma(-1.0, log((-1.0 / i)), log((-1.0 / n)))))));
	} else if (i <= 0.000155) {
		tmp = fma(100.0, (n * (0.3333333333333333 * (i * ((i / n) / n)))), (100.0 * (n + (n * (i * (0.5 - (0.5 / n)))))));
	} else {
		tmp = (n / i) * (100.0 * (n * (log(i) - log(n))));
	}
	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)
	tmp = 0.0
	if (i <= -5.1e+33)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (i <= -3.1e-53)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(Float64(n * fma(-1.0, log(Float64(-1.0 / i)), log(Float64(-1.0 / n))))))));
	elseif (i <= 0.000155)
		tmp = fma(100.0, Float64(n * Float64(0.3333333333333333 * Float64(i * Float64(Float64(i / n) / n)))), Float64(100.0 * Float64(n + Float64(n * Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	else
		tmp = Float64(Float64(n / i) * Float64(100.0 * Float64(n * Float64(log(i) - log(n)))));
	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_] := If[LessEqual[i, -5.1e+33], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -3.1e-53], N[(100.0 * N[(n / N[(i / N[(Exp[N[(n * N[(-1.0 * N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 0.000155], N[(100.0 * N[(n * N[(0.3333333333333333 * N[(i * N[(N[(i / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(100.0 * N[(n + N[(n * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * N[(100.0 * N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	26.6%
Target	26.1%
Herbie	77.6%

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

Derivation

1. Split input into 4 regimes

2. if i < -5.0999999999999999e33

   1. Initial program 62.2%

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

   2. Simplified 61.5%

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

      [Start] 62.2	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 61.5	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 61.5	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 61.5	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 61.5	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 61.5	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 61.5	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 61.5	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 82.3%

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

   4. Simplified 82.3%

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

      [Start] 82.3	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-/l* [=>] 82.3	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \]
      expm1-def [=>] 82.3	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   if -5.0999999999999999e33 < i < -3.10000000000000015e-53

   1. Initial program 19.3%

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

   2. Simplified 19.1%

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

      [Start] 19.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 19.1	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 19.1	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 19.1	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 19.1	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 19.1	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 19.1	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 19.1	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in i around -inf 15.4%

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

   4. Simplified 29.3%

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

      [Start] 15.4	\[ 100 \cdot \frac{n \cdot \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1\right)}{i} \]
      associate-/l* [=>] 15.4	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1}}} \]
      expm1-def [=>] 29.3	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)\right)}}} \]
      fma-def [=>] 29.3	\[ 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{fma}\left(-1, \log \left(\frac{-1}{i}\right), \log \left(-\frac{1}{n}\right)\right)}\right)}} \]
      distribute-neg-frac [=>] 29.3	\[ 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{i}\right), \log \color{blue}{\left(\frac{-1}{n}\right)}\right)\right)}} \]
      metadata-eval [=>] 29.3	\[ 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{i}\right), \log \left(\frac{\color{blue}{-1}}{n}\right)\right)\right)}} \]

   if -3.10000000000000015e-53 < i < 1.55e-4

   1. Initial program 9.1%

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

   2. Simplified 9.1%

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

      [Start] 9.1	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 9.1	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      associate-/l* [<=] 9.7	\[ \color{blue}{\frac{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}{i}} \]
      *-commutative [=>] 9.7	\[ \frac{\color{blue}{n \cdot \left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}}{i} \]
      associate-/l* [=>] 9.7	\[ \color{blue}{\frac{n}{\frac{i}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}} \]
      associate-/r/ [=>] 9.1	\[ \color{blue}{\frac{n}{i} \cdot \left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)} \]
      sub-neg [=>] 9.1	\[ \frac{n}{i} \cdot \left(100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}\right) \]
      distribute-lft-in [=>] 9.1	\[ \frac{n}{i} \cdot \color{blue}{\left(100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)\right)} \]
      fma-def [=>] 9.1	\[ \frac{n}{i} \cdot \color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)} \]
      metadata-eval [=>] 9.1	\[ \frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right) \]
      metadata-eval [=>] 9.1	\[ \frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right) \]

   3. Taylor expanded in i around 0 79.1%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) + \left(100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right)} \]

   4. Simplified 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right)} \]
      Proof

      [Start] 79.1	\[ 100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) + \left(100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      fma-def [=>] 79.1	\[ \color{blue}{\mathsf{fma}\left(100, n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right)} \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      associate--l+ [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      associate-*r/ [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      metadata-eval [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      associate-*r/ [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      metadata-eval [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      distribute-lft-out [=>] 79.1	\[ \mathsf{fma}\left(100, n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right), \color{blue}{100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n\right)}\right) \]

   5. Taylor expanded in n around 0 79.5%

      \[\leadsto \mathsf{fma}\left(100, n \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{i}^{2}}{{n}^{2}}\right)}, 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]

   6. Simplified 86.0%

      \[\leadsto \mathsf{fma}\left(100, n \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\frac{\frac{i}{n}}{n} \cdot i\right)\right)}, 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]
      Proof

      [Start] 79.5	\[ \mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \frac{{i}^{2}}{{n}^{2}}\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]
      unpow2 [=>] 79.5	\[ \mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{i \cdot i}}{{n}^{2}}\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]
      unpow2 [=>] 79.5	\[ \mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \frac{i \cdot i}{\color{blue}{n \cdot n}}\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]
      associate-*l/ [<=] 79.5	\[ \mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{i}{n \cdot n} \cdot i\right)}\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]
      associate-/r* [=>] 86.0	\[ \mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\frac{\frac{i}{n}}{n}} \cdot i\right)\right), 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) + n\right)\right) \]

   if 1.55e-4 < i

   1. Initial program 50.8%

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

   2. Simplified 50.8%

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

      [Start] 50.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 50.8	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      associate-/l* [<=] 50.8	\[ \color{blue}{\frac{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}{i}} \]
      *-commutative [=>] 50.8	\[ \frac{\color{blue}{n \cdot \left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}}{i} \]
      associate-/l* [=>] 50.8	\[ \color{blue}{\frac{n}{\frac{i}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}} \]
      associate-/r/ [=>] 50.8	\[ \color{blue}{\frac{n}{i} \cdot \left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)} \]
      sub-neg [=>] 50.8	\[ \frac{n}{i} \cdot \left(100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}\right) \]
      distribute-lft-in [=>] 50.8	\[ \frac{n}{i} \cdot \color{blue}{\left(100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)\right)} \]
      fma-def [=>] 50.8	\[ \frac{n}{i} \cdot \color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)} \]
      metadata-eval [=>] 50.8	\[ \frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right) \]
      metadata-eval [=>] 50.8	\[ \frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right) \]

   3. Taylor expanded in n around 0 64.0%

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

   4. Simplified 64.0%

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

      [Start] 64.0	\[ \frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(-1 \cdot \log n + \log i\right)\right)\right) \]
      +-commutative [=>] 64.0	\[ \frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \color{blue}{\left(\log i + -1 \cdot \log n\right)}\right)\right) \]
      mul-1-neg [=>] 64.0	\[ \frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i + \color{blue}{\left(-\log n\right)}\right)\right)\right) \]
      unsub-neg [=>] 64.0	\[ \frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \color{blue}{\left(\log i - \log n\right)}\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{i}\right), \log \left(\frac{-1}{n}\right)\right)\right)}}\\ \mathbf{elif}\;i \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.6%
Cost	20424

\[\begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 0.000118:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	77.6%
Cost	20424

\[\begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	77.5%
Cost	13900

\[\begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot \log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;i \leq 0.000145:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right)\\ \end{array} \]

Alternative 4
Accuracy	77.5%
Cost	13900

\[\begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot \log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;i \leq 0.00012:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	76.2%
Cost	8396

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot t_0}}\\ \mathbf{elif}\;i \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(100, n \cdot \left(0.3333333333333333 \cdot \left(i \cdot \frac{\frac{i}{n}}{n}\right)\right), 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 7.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{100 \cdot \left(t_0 \cdot \left(n \cdot n\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	76.3%
Cost	8140

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -5.2 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot t_0}}\\ \mathbf{elif}\;i \leq 0.000145:\\ \;\;\;\;\mathsf{fma}\left(100, \frac{0.3333333333333333 \cdot \left(i \cdot i\right)}{n}, 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{100 \cdot \left(t_0 \cdot \left(n \cdot n\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	76.1%
Cost	7632

\[\begin{array}{l} t_0 := 100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log \left(\frac{i}{n}\right)}{i}\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 0.00014:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + \left(\left(1 + i \cdot 0.5\right) + \left(-1 - i \cdot \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.75 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	76.1%
Cost	7632

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{t_0}{i}\right)\\ \mathbf{elif}\;i \leq 0.00015:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + \left(\left(1 + i \cdot 0.5\right) + \left(-1 - i \cdot \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{+111}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	76.2%
Cost	7632

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot t_0}}\\ \mathbf{elif}\;i \leq 0.00015:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + \left(\left(1 + i \cdot 0.5\right) + \left(-1 - i \cdot \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.55 \cdot 10^{+124}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	76.1%
Cost	7632

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -6.2 \cdot 10^{+36}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot t_0}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + \left(\left(1 + i \cdot 0.5\right) + \left(-1 - i \cdot \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+115}:\\ \;\;\;\;100 \cdot \frac{t_0}{\frac{i}{n \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	76.2%
Cost	7632

\[\begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{n \cdot t_0}}\\ \mathbf{elif}\;i \leq 0.00012:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + \left(\left(1 + i \cdot 0.5\right) + \left(-1 - i \cdot \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{100 \cdot \left(t_0 \cdot \left(n \cdot n\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	79.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 500:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	79.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 470:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	64.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.38:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Accuracy	67.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 1.7:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Accuracy	64.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.022:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 17
Accuracy	20.7%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023126 
(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))))