Compound Interest

Average Error: 47.2 → 12.9

Time: 35.2s

Precision: binary64

Cost: 39812

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0125:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(\mathsf{fma}\left(-\log \left(\frac{-1}{i}\right) \cdot \sqrt[3]{n}, \sqrt[3]{n \cdot n}, n \cdot \log \left(\frac{-1}{n}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.019:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \left(\left(-\log \left(\frac{1}{i}\right)\right) + \log \left(\frac{1}{n}\right)\right)\right)}{\frac{i}{n}}\\ \end{array} \]

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 -0.0125)
   (*
    100.0
    (/
     (expm1
      (fma
       (- (* (log (/ -1.0 i)) (cbrt n)))
       (cbrt (* n n))
       (* n (log (/ -1.0 n)))))
     (/ i n)))
   (if (<= i 0.019)
     (* 100.0 n)
     (*
      100.0
      (/ (expm1 (* n (+ (- (log (/ 1.0 i))) (log (/ 1.0 n))))) (/ i 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 <= -0.0125) {
		tmp = 100.0 * (expm1(fma(-(log((-1.0 / i)) * cbrt(n)), cbrt((n * n)), (n * log((-1.0 / n))))) / (i / n));
	} else if (i <= 0.019) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (expm1((n * (-log((1.0 / i)) + log((1.0 / n))))) / (i / 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 <= -0.0125)
		tmp = Float64(100.0 * Float64(expm1(fma(Float64(-Float64(log(Float64(-1.0 / i)) * cbrt(n))), cbrt(Float64(n * n)), Float64(n * log(Float64(-1.0 / n))))) / Float64(i / n)));
	elseif (i <= 0.019)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(expm1(Float64(n * Float64(Float64(-log(Float64(1.0 / i))) + log(Float64(1.0 / n))))) / Float64(i / 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, -0.0125], N[(100.0 * N[(N[(Exp[N[((-N[(N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision] * N[Power[n, 1/3], $MachinePrecision]), $MachinePrecision]) * N[Power[N[(n * n), $MachinePrecision], 1/3], $MachinePrecision] + N[(n * N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 0.019], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(Exp[N[(n * N[((-N[Log[N[(1.0 / i), $MachinePrecision]], $MachinePrecision]) + N[Log[N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	47.2
Target	47.2
Herbie	12.9

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

Derivation

1. Split input into 3 regimes

2. if i < -0.012500000000000001

   1. Initial program 27.3

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

   2. Taylor expanded in i around -inf 30.1

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

   3. Simplified 25.1

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

   4. Applied egg-rr 16.7

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-\log \left(\frac{-1}{i}\right) \cdot \sqrt[3]{n}, \sqrt[3]{n \cdot n}, n \cdot \log \left(\frac{-1}{n}\right)\right)}\right)}{\frac{i}{n}} \]

   if -0.012500000000000001 < i < 0.0189999999999999995

   1. Initial program 57.9

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

   2. Taylor expanded in i around 0 10.0

      \[\leadsto 100 \cdot \color{blue}{n} \]

   if 0.0189999999999999995 < i

   1. Initial program 31.4

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

   2. Taylor expanded in i around inf 28.4

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

   3. Simplified 20.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \left(\left(-\log \left(\frac{1}{i}\right)\right) + \log \left(\frac{1}{n}\right)\right)\right)}}{\frac{i}{n}} \]
      Proof
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	13.1
Cost	20488

\[\begin{array}{l} \mathbf{if}\;i \leq -0.475:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.245:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \left(\left(-\log \left(\frac{1}{i}\right)\right) + \log \left(\frac{1}{n}\right)\right)\right)}{\frac{i}{n}}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0062:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 92:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot n\right)\\ \end{array} \]

Alternative 3
Error	22.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+40}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;n \leq -2.1 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-63}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]

Alternative 4
Error	20.0
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \left(\frac{n}{i} \cdot n\right)\\ \mathbf{if}\;i \leq -0.0078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 0.175:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	28.5
Cost	192

\[100 \cdot n \]

Reproduce

herbie shell --seed 2023010 
(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))))