Compound Interest

Average Error: 47.8 → 0.5

Time: 22.5s

Precision: binary64

Cost: 28040

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 200000:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(100, t_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \end{array} \]

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 (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 200000.0)
       (/ (* n (fma 100.0 t_0 -100.0)) i)
       (+ (+ 1.0 (* n 100.0)) -1.0)))))

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 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= 200000.0) {
		tmp = (n * fma(100.0, t_0, -100.0)) / i;
	} else {
		tmp = (1.0 + (n * 100.0)) + -1.0;
	}
	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(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= 200000.0)
		tmp = Float64(Float64(n * fma(100.0, t_0, -100.0)) / i);
	else
		tmp = Float64(Float64(1.0 + Float64(n * 100.0)) + -1.0);
	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[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 200000.0], N[(N[(n * N[(100.0 * t$95$0 + -100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(N[(1.0 + N[(n * 100.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Target

Original	47.8
Target	47.6
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 46.2

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

   2. Applied egg-rr 0.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e5

   1. Initial program 2.7

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{\frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)} \]
      Proof
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) -100)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (*.f64 -1 100)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n)) (*.f64 (neg.f64 1) 100)))): 3 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 n i) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 100)) (*.f64 (neg.f64 1) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= distribute-rgt-in_binary64 (*.f64 100 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (neg.f64 1))))): 3 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 n i) (*.f64 100 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 i (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))))): 3 points increase in error, 54 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))) i)): 4 points increase in error, 4 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 n (/.f64 (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 5 points increase in error, 2 points decrease in error
      (*.f64 n (Rewrite<= associate-*r/_binary64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 2 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)) n)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 100 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 1 points increase in error, 5 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 40 points increase in error, 2 points decrease in error

   3. Applied egg-rr 2.6

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

   if 2e5 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 62.9

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

   2. Taylor expanded in i around 0 49.8

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

   3. Simplified 49.8

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]
      Proof
      (+.f64 i 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 i)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 14.5

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

   5. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\left(1 + n \cdot 100\right) - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 200000:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	21768

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_0 \leq 200000:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \end{array} \]

Alternative 2
Error	1.2
Cost	21768

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_0 \leq 200000:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	21768

\[\begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_0 \leq 200000:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \end{array} \]

Alternative 4
Error	12.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq 4.2 \cdot 10^{-133}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternative 5
Error	22.4
Cost	712

\[\begin{array}{l} t_0 := i \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	22.4
Cost	712

\[\begin{array}{l} t_0 := i \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.4
Cost	712

\[\begin{array}{l} t_0 := \left(1 + n \cdot 100\right) + -1\\ \mathbf{if}\;i \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{-133}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	19.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;\left(1 + n \cdot 100\right) + -1\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{-133}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(1 + n\right) + -1\right)\\ \end{array} \]

Alternative 9
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 10
Error	28.1
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2022294 
(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))))