Compound Interest

Average Error: 47.2 → 12.0

Time: 10.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{n \cdot n}{i}\\ \mathbf{if}\;i \leq -8.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-50, \mathsf{fma}\left(i, i, i\right), \mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, \mathsf{fma}\left(i, i \cdot 16.666666666666668, 100\right)\right), i \cdot \frac{i}{\frac{n}{33.333333333333336}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{fma}\left(100, \frac{n}{i} \cdot \log i, \mathsf{fma}\left(50, \frac{n \cdot n}{\frac{i}{{\log i}^{2}}}, \mathsf{fma}\left(100, \frac{n}{i} \cdot \frac{n}{i}, 50 \cdot \left(t_0 \cdot {\log n}^{2}\right)\right)\right)\right) + -100 \cdot \left(t_0 \cdot \left(\log i \cdot \log n\right) + \frac{n}{i} \cdot \log n\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* n n) i)))
   (if (<= i -8.5e-22)
     (/ (fma (exp i) 100.0 -100.0) (/ i n))
     (if (<= i 2.7e-8)
       (fma
        -50.0
        (fma i i i)
        (fma
         n
         (fma i 50.0 (fma i (* i 16.666666666666668) 100.0))
         (* i (/ i (/ n 33.333333333333336)))))
       (*
        n
        (+
         (fma
          100.0
          (* (/ n i) (log i))
          (fma
           50.0
           (/ (* n n) (/ i (pow (log i) 2.0)))
           (fma 100.0 (* (/ n i) (/ n i)) (* 50.0 (* t_0 (pow (log n) 2.0))))))
         (* -100.0 (+ (* t_0 (* (log i) (log n))) (* (/ n 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 t_0 = (n * n) / i;
	double tmp;
	if (i <= -8.5e-22) {
		tmp = fma(exp(i), 100.0, -100.0) / (i / n);
	} else if (i <= 2.7e-8) {
		tmp = fma(-50.0, fma(i, i, i), fma(n, fma(i, 50.0, fma(i, (i * 16.666666666666668), 100.0)), (i * (i / (n / 33.333333333333336)))));
	} else {
		tmp = n * (fma(100.0, ((n / i) * log(i)), fma(50.0, ((n * n) / (i / pow(log(i), 2.0))), fma(100.0, ((n / i) * (n / i)), (50.0 * (t_0 * pow(log(n), 2.0)))))) + (-100.0 * ((t_0 * (log(i) * log(n))) + ((n / 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)
	t_0 = Float64(Float64(n * n) / i)
	tmp = 0.0
	if (i <= -8.5e-22)
		tmp = Float64(fma(exp(i), 100.0, -100.0) / Float64(i / n));
	elseif (i <= 2.7e-8)
		tmp = fma(-50.0, fma(i, i, i), fma(n, fma(i, 50.0, fma(i, Float64(i * 16.666666666666668), 100.0)), Float64(i * Float64(i / Float64(n / 33.333333333333336)))));
	else
		tmp = Float64(n * Float64(fma(100.0, Float64(Float64(n / i) * log(i)), fma(50.0, Float64(Float64(n * n) / Float64(i / (log(i) ^ 2.0))), fma(100.0, Float64(Float64(n / i) * Float64(n / i)), Float64(50.0 * Float64(t_0 * (log(n) ^ 2.0)))))) + Float64(-100.0 * Float64(Float64(t_0 * Float64(log(i) * log(n))) + Float64(Float64(n / 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_] := Block[{t$95$0 = N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -8.5e-22], N[(N[(N[Exp[i], $MachinePrecision] * 100.0 + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.7e-8], N[(-50.0 * N[(i * i + i), $MachinePrecision] + N[(n * N[(i * 50.0 + N[(i * N[(i * 16.666666666666668), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision] + N[(i * N[(i / N[(n / 33.333333333333336), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[Log[i], $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(N[(n * n), $MachinePrecision] / N[(i / N[Power[N[Log[i], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(t$95$0 * N[Power[N[Log[n], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-100.0 * N[(N[(t$95$0 * N[(N[Log[i], $MachinePrecision] * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n / i), $MachinePrecision] * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus i

Bits error versus n

Target

Original	47.2
Target	47.3
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if i < -8.5000000000000001e-22

   1. Initial program 29.2

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

   2. Simplified 29.7

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

   3. Taylor expanded in n around inf 14.8

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

   4. Simplified 14.1

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

   if -8.5000000000000001e-22 < i < 2.70000000000000002e-8

   1. Initial program 58.3

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

   2. Simplified 57.9

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

   3. Taylor expanded in i around 0 13.0

      \[\leadsto n \cdot \color{blue}{\left(\left(100 + \left(50 \cdot i + \left(16.666666666666668 \cdot {i}^{2} + 33.333333333333336 \cdot \frac{{i}^{2}}{{n}^{2}}\right)\right)\right) - \left(50 \cdot \frac{i}{n} + 50 \cdot \frac{{i}^{2}}{n}\right)\right)} \]

   4. Simplified 8.8

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(\mathsf{fma}\left(50, i, \mathsf{fma}\left(16.666666666666668, i \cdot i, 33.333333333333336 \cdot \left(\frac{i}{n} \cdot \frac{i}{n}\right)\right)\right) + -50 \cdot \left(\frac{i}{n} + \frac{i \cdot i}{n}\right)\right)\right)} \]

   5. Taylor expanded in n around 0 8.9

      \[\leadsto \color{blue}{\left(100 \cdot n + \left(16.666666666666668 \cdot \left(n \cdot {i}^{2}\right) + \left(33.333333333333336 \cdot \frac{{i}^{2}}{n} + 50 \cdot \left(n \cdot i\right)\right)\right)\right) - \left(50 \cdot i + 50 \cdot {i}^{2}\right)} \]

   6. Simplified 8.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(-50, \mathsf{fma}\left(i, i, i\right), \mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, \mathsf{fma}\left(i, i \cdot 16.666666666666668, 100\right)\right), \frac{i}{\frac{n}{33.333333333333336}} \cdot i\right)\right)} \]

   if 2.70000000000000002e-8 < i

   1. Initial program 32.2

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

   2. Simplified 32.2

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

   3. Taylor expanded in n around 0 21.8

      \[\leadsto n \cdot \color{blue}{\left(\left(100 \cdot \frac{n \cdot \log i}{i} + \left(50 \cdot \frac{{\log i}^{2} \cdot {n}^{2}}{i} + \left(100 \cdot \frac{{n}^{2}}{{i}^{2}} + 50 \cdot \frac{{n}^{2} \cdot {\log n}^{2}}{i}\right)\right)\right) - \left(100 \cdot \frac{\log i \cdot \left({n}^{2} \cdot \log n\right)}{i} + 100 \cdot \frac{n \cdot \log n}{i}\right)\right)} \]

   4. Simplified 21.8

      \[\leadsto n \cdot \color{blue}{\left(\mathsf{fma}\left(100, \frac{n}{i} \cdot \log i, \mathsf{fma}\left(50, \frac{n \cdot n}{\frac{i}{{\log i}^{2}}}, \mathsf{fma}\left(100, \frac{n}{i} \cdot \frac{n}{i}, 50 \cdot \left(\frac{n \cdot n}{i} \cdot {\log n}^{2}\right)\right)\right)\right) + -100 \cdot \left(\frac{n \cdot n}{i} \cdot \left(\log i \cdot \log n\right) + \frac{n}{i} \cdot \log n\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-50, \mathsf{fma}\left(i, i, i\right), \mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, \mathsf{fma}\left(i, i \cdot 16.666666666666668, 100\right)\right), i \cdot \frac{i}{\frac{n}{33.333333333333336}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{fma}\left(100, \frac{n}{i} \cdot \log i, \mathsf{fma}\left(50, \frac{n \cdot n}{\frac{i}{{\log i}^{2}}}, \mathsf{fma}\left(100, \frac{n}{i} \cdot \frac{n}{i}, 50 \cdot \left(\frac{n \cdot n}{i} \cdot {\log n}^{2}\right)\right)\right)\right) + -100 \cdot \left(\frac{n \cdot n}{i} \cdot \left(\log i \cdot \log n\right) + \frac{n}{i} \cdot \log n\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))