Compound Interest

Average Accuracy: 26.0% → 81.2%

Time: 27.2s

Precision: binary64

Cost: 46984

Math

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

↓

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ t_1 := \log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{t_1} + \mathsf{expm1}\left(n \cdot t_1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(-1, \log n, \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* 100.0 (* n (/ (expm1 i) i))))
        (t_1 (- (log (/ -1.0 n)) (log (/ -1.0 i)))))
   (if (<= n -3.8e-125)
     t_0
     (if (<= n -4e-310)
       (/
        (* 100.0 (+ (* (/ (* n n) i) (pow (exp n) t_1)) (expm1 (* n t_1))))
        (/ i n))
       (if (<= n 2.1e-141)
         (/ (* (* n 100.0) (fma -1.0 (log n) (log i))) (/ i n))
         (if (<= n 5.3e-14) (/ (* 100.0 i) (/ i n)) t_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 = 100.0 * (n * (expm1(i) / i));
	double t_1 = log((-1.0 / n)) - log((-1.0 / i));
	double tmp;
	if (n <= -3.8e-125) {
		tmp = t_0;
	} else if (n <= -4e-310) {
		tmp = (100.0 * ((((n * n) / i) * pow(exp(n), t_1)) + expm1((n * t_1)))) / (i / n);
	} else if (n <= 2.1e-141) {
		tmp = ((n * 100.0) * fma(-1.0, log(n), log(i))) / (i / n);
	} else if (n <= 5.3e-14) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_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(100.0 * Float64(n * Float64(expm1(i) / i)))
	t_1 = Float64(log(Float64(-1.0 / n)) - log(Float64(-1.0 / i)))
	tmp = 0.0
	if (n <= -3.8e-125)
		tmp = t_0;
	elseif (n <= -4e-310)
		tmp = Float64(Float64(100.0 * Float64(Float64(Float64(Float64(n * n) / i) * (exp(n) ^ t_1)) + expm1(Float64(n * t_1)))) / Float64(i / n));
	elseif (n <= 2.1e-141)
		tmp = Float64(Float64(Float64(n * 100.0) * fma(-1.0, log(n), log(i))) / Float64(i / n));
	elseif (n <= 5.3e-14)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = t_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[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.8e-125], t$95$0, If[LessEqual[n, -4e-310], N[(N[(100.0 * N[(N[(N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision] * N[Power[N[Exp[n], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[(n * t$95$1), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-141], N[(N[(N[(n * 100.0), $MachinePrecision] * N[(-1.0 * N[Log[n], $MachinePrecision] + N[Log[i], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-14], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Target

Original	26.0%
Target	26.2%
Herbie	81.2%

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

Derivation

1. Split input into 4 regimes

2. if n < -3.8000000000000001e-125 or 5.3000000000000001e-14 < n

   1. Initial program 19.6%

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

   2. Simplified 20.0%

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

      [Start] 19.6	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 20.0	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 20.0	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 20.0	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 20.0	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 20.0	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 20.0	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 20.0	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 25.9%

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

   4. Simplified 87.3%

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

      [Start] 25.9	\[ 100 \cdot \left(n \cdot \frac{e^{i} - 1}{i}\right) \]
      expm1-def [=>] 87.3	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \]

   if -3.8000000000000001e-125 < n < -3.999999999999988e-310

   1. Initial program 62.0%

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

   2. Simplified 62.0%

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

      [Start] 62.0	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 62.0	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 62.0	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 62.0	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around -inf 61.2%

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

   4. Simplified 72.2%

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

      [Start] 61.2	\[ \frac{100 \cdot \left(\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}}{i} + e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}\right) - 1\right)}{\frac{i}{n}} \]
      associate--l+ [=>] 61.2	\[ \frac{100 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}}{i} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1\right)\right)}}{\frac{i}{n}} \]

   if -3.999999999999988e-310 < n < 2.0999999999999999e-141

   1. Initial program 34.6%

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

   2. Simplified 34.6%

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

      [Start] 34.6	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 34.6	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 34.6	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 34.6	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in n around 0 75.8%

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

   4. Simplified 75.8%

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

      [Start] 75.8	\[ \frac{100 \cdot \left(n \cdot \left(-1 \cdot \log n + \log i\right)\right)}{\frac{i}{n}} \]
      associate-*r* [=>] 75.8	\[ \frac{\color{blue}{\left(100 \cdot n\right) \cdot \left(-1 \cdot \log n + \log i\right)}}{\frac{i}{n}} \]
      fma-def [=>] 75.8	\[ \frac{\left(100 \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \log n, \log i\right)}}{\frac{i}{n}} \]

   if 2.0999999999999999e-141 < n < 5.3000000000000001e-14

   1. Initial program 14.1%

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

   2. Simplified 14.1%

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

      [Start] 14.1	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 14.1	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 14.1	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 14.1	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around 0 59.9%

      \[\leadsto \frac{\color{blue}{100 \cdot i}}{\frac{i}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-125}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)} + \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(-1, \log n, \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.2%
Cost	20424

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -4 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(-1, \log n, \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	81.3%
Cost	20424

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(-1, \log n, \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	81.2%
Cost	20300

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(\log \left(-i\right) - \log \left(-n\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(-1, \log n, \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	79.4%
Cost	14032

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.05 \cdot 10^{-149}:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right)\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	79.9%
Cost	14032

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.58 \cdot 10^{-149}:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	79.4%
Cost	14032

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	81.2%
Cost	13900

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(\log \left(-i\right) - \log \left(-n\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)\right)\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	79.7%
Cost	7508

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ t_1 := \frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -2.3 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-149}:\\ \;\;\;\;\log \left(\frac{i}{n}\right) \cdot \left(n \cdot \frac{n \cdot 100}{i}\right)\\ \mathbf{elif}\;n \leq -9.4 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	79.6%
Cost	7508

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ t_1 := \frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -6.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq -1.08 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	80.1%
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-198}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	67.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;i \leq -2.3 \cdot 10^{+23} \lor \neg \left(i \leq 5200000000\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + n \cdot \left(i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	67.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;i \leq -2.3 \cdot 10^{+21} \lor \neg \left(i \leq 5200000000\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	64.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \lor \neg \left(i \leq 240\right):\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(n \cdot i\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 14
Accuracy	67.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -0.9 \lor \neg \left(i \leq 5200000000\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(n \cdot i\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 15
Accuracy	63.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \lor \neg \left(i \leq 128\right):\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 16
Accuracy	64.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.45 \lor \neg \left(i \leq 250\right):\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 17
Accuracy	56.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-250}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-224}:\\ \;\;\;\;50 \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 18
Accuracy	3.0%
Cost	192

\[i \cdot -50 \]

Alternative 19
Accuracy	55.4%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023133 
(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))))