Compound Interest

Average Error: 47.7 → 13.9

Time: 10.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ \mathbf{if}\;i \leq -5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot t_0\right) + -100 \cdot n}{i}\\ \mathbf{elif}\;i \leq -4.3 \cdot 10^{-24}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{i}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.076:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, i, i\right), -50, \mathsf{fma}\left(16.666666666666668, i \cdot \left(i \cdot n\right), \mathsf{fma}\left(100, n, \mathsf{fma}\left(33.333333333333336, \frac{i}{\frac{n}{i}}, 50 \cdot \left(i \cdot n\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(t_0, 100, -100\right)}{i}\\ \end{array} \]

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 (+ (/ i n) 1.0) n)))
   (if (<= i -5e+100)
     (/ (fma (exp i) 100.0 -100.0) (/ i n))
     (if (<= i -9.5e+68)
       (/ (+ (* n (* 100.0 t_0)) (* -100.0 n)) i)
       (if (<= i -4.3e-24)
         (* n (/ (fma 100.0 (exp i) -100.0) i))
         (if (<= i 0.076)
           (fma
            (fma i i i)
            -50.0
            (fma
             16.666666666666668
             (* i (* i n))
             (fma
              100.0
              n
              (fma 33.333333333333336 (/ i (/ n i)) (* 50.0 (* i n))))))
           (/ (* n (fma t_0 100.0 -100.0)) i)))))))

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(((i / n) + 1.0), n);
	double tmp;
	if (i <= -5e+100) {
		tmp = fma(exp(i), 100.0, -100.0) / (i / n);
	} else if (i <= -9.5e+68) {
		tmp = ((n * (100.0 * t_0)) + (-100.0 * n)) / i;
	} else if (i <= -4.3e-24) {
		tmp = n * (fma(100.0, exp(i), -100.0) / i);
	} else if (i <= 0.076) {
		tmp = fma(fma(i, i, i), -50.0, fma(16.666666666666668, (i * (i * n)), fma(100.0, n, fma(33.333333333333336, (i / (n / i)), (50.0 * (i * n))))));
	} else {
		tmp = (n * fma(t_0, 100.0, -100.0)) / i;
	}
	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(i / n) + 1.0) ^ n
	tmp = 0.0
	if (i <= -5e+100)
		tmp = Float64(fma(exp(i), 100.0, -100.0) / Float64(i / n));
	elseif (i <= -9.5e+68)
		tmp = Float64(Float64(Float64(n * Float64(100.0 * t_0)) + Float64(-100.0 * n)) / i);
	elseif (i <= -4.3e-24)
		tmp = Float64(n * Float64(fma(100.0, exp(i), -100.0) / i));
	elseif (i <= 0.076)
		tmp = fma(fma(i, i, i), -50.0, fma(16.666666666666668, Float64(i * Float64(i * n)), fma(100.0, n, fma(33.333333333333336, Float64(i / Float64(n / i)), Float64(50.0 * Float64(i * n))))));
	else
		tmp = Float64(Float64(n * fma(t_0, 100.0, -100.0)) / i);
	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[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, If[LessEqual[i, -5e+100], N[(N[(N[Exp[i], $MachinePrecision] * 100.0 + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -9.5e+68], N[(N[(N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, -4.3e-24], N[(n * N[(N[(100.0 * N[Exp[i], $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 0.076], N[(N[(i * i + i), $MachinePrecision] * -50.0 + N[(16.666666666666668 * N[(i * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(100.0 * n + N[(33.333333333333336 * N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]

Error

Bits error versus i

Bits error versus n

Target

Original	47.7
Target	47.5
Herbie	13.9

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

Derivation

1. Split input into 5 regimes

2. if i < -4.9999999999999999e100

   1. Initial program 19.8

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

   2. Simplified 20.3

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

   3. Taylor expanded in n around inf 9.5

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

   4. Simplified 8.7

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

   if -4.9999999999999999e100 < i < -9.50000000000000069e68

   1. Initial program 36.1

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

   2. Simplified 36.6

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

   3. Applied egg-rr 36.6

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

   4. Applied egg-rr 36.6

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

   if -9.50000000000000069e68 < i < -4.3000000000000003e-24

   1. Initial program 45.5

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

   2. Simplified 45.7

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

   3. Applied egg-rr 45.7

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

   4. Taylor expanded in n around inf 26.4

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

   5. Simplified 26.2

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

   if -4.3000000000000003e-24 < i < 0.0759999999999999981

   1. Initial program 58.6

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

   2. Simplified 58.2

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

   3. Taylor expanded in i around 0 8.6

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

   4. Simplified 8.6

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

   5. Applied egg-rr 8.5

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

   if 0.0759999999999999981 < i

   1. Initial program 31.9

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

   2. Simplified 31.9

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

   3. Applied egg-rr 31.9

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

   4. Applied egg-rr 31.9

      \[\leadsto \frac{n \cdot \mathsf{fma}\left(\color{blue}{1 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}, 100, -100\right)}{i} \]
3. Recombined 5 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) + -100 \cdot n}{i}\\ \mathbf{elif}\;i \leq -4.3 \cdot 10^{-24}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{i}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.076:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, i, i\right), -50, \mathsf{fma}\left(16.666666666666668, i \cdot \left(i \cdot n\right), \mathsf{fma}\left(100, n, \mathsf{fma}\left(33.333333333333336, \frac{i}{\frac{n}{i}}, 50 \cdot \left(i \cdot n\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, 100, -100\right)}{i}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))