Compound Interest

Average Error: 48.0 → 10.7

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0005:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 5.9 \cdot 10^{+35}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{n}{i} \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

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 -0.0005)
   (* n (/ (fma 100.0 (exp (* n (log1p (/ i n)))) -100.0) i))
   (if (<= i 5.9e+35)
     (+ (* 100.0 (* n (* i (+ 0.5 (* 0.5 (/ -1.0 n)))))) (* n 100.0))
     (* n (* 100.0 (* (/ n i) (- (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 <= -0.0005) {
		tmp = n * (fma(100.0, exp((n * log1p((i / n)))), -100.0) / i);
	} else if (i <= 5.9e+35) {
		tmp = (100.0 * (n * (i * (0.5 + (0.5 * (-1.0 / n)))))) + (n * 100.0);
	} else {
		tmp = n * (100.0 * ((n / i) * (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 <= -0.0005)
		tmp = Float64(n * Float64(fma(100.0, exp(Float64(n * log1p(Float64(i / n)))), -100.0) / i));
	elseif (i <= 5.9e+35)
		tmp = Float64(Float64(100.0 * Float64(n * Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))))) + Float64(n * 100.0));
	else
		tmp = Float64(n * Float64(100.0 * Float64(Float64(n / i) * 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, -0.0005], N[(n * N[(N[(100.0 * N[Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.9e+35], N[(N[(100.0 * N[(n * N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	48.0
Target	47.9
Herbie	10.7

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

Derivation

1. Split input into 3 regimes

2. if i < -5.0000000000000001e-4

   1. Initial program 27.6

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

   2. Simplified 28.1

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

   3. Applied egg-rr 6.0

      \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, \color{blue}{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{i} \]

   if -5.0000000000000001e-4 < i < 5.89999999999999985e35

   1. Initial program 57.9

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

   2. Simplified 57.5

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

   3. Taylor expanded in i around 0 10.3

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

   if 5.89999999999999985e35 < i

   1. Initial program 32.8

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

   2. Simplified 32.8

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

   3. Taylor expanded in n around 0 21.5

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

   4. Simplified 21.6

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0005:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 5.9 \cdot 10^{+35}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right) + n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{n}{i} \cdot \left(\log i - \log n\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))