Compound Interest

Percentage Accurate: 27.7% → 94.6%

Time: 15.1s

Alternatives: 13

Speedup: 12.2×

• could not determine a ground truth

  1. i = 3.0863771931507095e-108
  2. n = -2.4034732482253767e+39

Specification

Math

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

FPCore

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

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
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]

Initial Program: 27.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
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]

Alternative 1: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-215}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(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 2e-215)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY) (/ (* n (fma t_0 100.0 -100.0)) i) (* n 100.0)))))

C

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 <= 2e-215) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * fma(t_0, 100.0, -100.0)) / i;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Julia

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 <= 2e-215)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * fma(t_0, 100.0, -100.0)) / i);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

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, 2e-215], 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, Infinity], N[(N[(n * N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 2.00000000000000008e-215

   1. Initial program 21.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   if 2.00000000000000008e-215 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*r* N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 80.2

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

   5. Simplified 80.2%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-215}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{if}\;i \leq -6 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right) + \frac{\frac{\left(i \cdot i\right) \cdot 0.3333333333333333}{n} - i \cdot \mathsf{fma}\left(i, 0.5, 0.5\right)}{n}\right)\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{1}{i}, n \cdot {\left(\frac{i}{n}\right)}^{n}, \frac{n}{-i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ 100.0 i) (* n (expm1 (* n (log1p (/ i n))))))))
   (if (<= i -6e-95)
     t_0
     (if (<= i 2.8e-155)
       (*
        100.0
        (*
         n
         (+
          (fma i (fma i 0.16666666666666666 0.5) 1.0)
          (/
           (- (/ (* (* i i) 0.3333333333333333) n) (* i (fma i 0.5 0.5)))
           n))))
       (if (<= i 3.1e+117)
         t_0
         (* 100.0 (fma (/ 1.0 i) (* n (pow (/ i n) n)) (/ n (- i)))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 / i) * (n * expm1((n * log1p((i / n)))));
	double tmp;
	if (i <= -6e-95) {
		tmp = t_0;
	} else if (i <= 2.8e-155) {
		tmp = 100.0 * (n * (fma(i, fma(i, 0.16666666666666666, 0.5), 1.0) + (((((i * i) * 0.3333333333333333) / n) - (i * fma(i, 0.5, 0.5))) / n)));
	} else if (i <= 3.1e+117) {
		tmp = t_0;
	} else {
		tmp = 100.0 * fma((1.0 / i), (n * pow((i / n), n)), (n / -i));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 / i) * Float64(n * expm1(Float64(n * log1p(Float64(i / n))))))
	tmp = 0.0
	if (i <= -6e-95)
		tmp = t_0;
	elseif (i <= 2.8e-155)
		tmp = Float64(100.0 * Float64(n * Float64(fma(i, fma(i, 0.16666666666666666, 0.5), 1.0) + Float64(Float64(Float64(Float64(Float64(i * i) * 0.3333333333333333) / n) - Float64(i * fma(i, 0.5, 0.5))) / n))));
	elseif (i <= 3.1e+117)
		tmp = t_0;
	else
		tmp = Float64(100.0 * fma(Float64(1.0 / i), Float64(n * (Float64(i / n) ^ n)), Float64(n / Float64(-i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 / i), $MachinePrecision] * N[(n * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -6e-95], t$95$0, If[LessEqual[i, 2.8e-155], N[(100.0 * N[(n * N[(N[(i * N[(i * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(N[(N[(N[(i * i), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] - N[(i * N[(i * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e+117], t$95$0, N[(100.0 * N[(N[(1.0 / i), $MachinePrecision] * N[(n * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] + N[(n / (-i)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -6e-95 or 2.8e-155 < i < 3.09999999999999975e117

   1. Initial program 35.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. div-inv N/A

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

      16. clear-num N/A

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

      17. /-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 35.2

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

   4. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{\frac{100}{i} \cdot \left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 N/A

         \[\leadsto \frac{100}{i} \cdot \left(n \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n} + -1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{100}{i} \cdot \left(n \cdot \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} + -1\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{100}{i} \cdot \left(n \cdot \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} + -1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{100}{i} \cdot \left(n \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{100}{i} \cdot \left(n \cdot \color{blue}{\left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1\right)}\right) \]

      10. lift-expm1.f64 88.6

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

   6. Applied egg-rr 88.6%

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

   if -6e-95 < i < 2.8e-155

   1. Initial program 5.6%

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

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + 1\right)} - 1}{\frac{i}{n}} \]

      2. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i, 1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right), 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 1.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, \frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right), 0.5 - \frac{0.5}{n}\right), 1\right), 1\right)} - 1}{\frac{i}{n}} \]

   6. Taylor expanded in n around -inf

      \[\leadsto 100 \cdot \color{blue}{\left(-1 \cdot \left(n \cdot \left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\mathsf{neg}\left(n \cdot \left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right) \cdot n}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right) \cdot \left(\mathsf{neg}\left(n\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto 100 \cdot \left(\left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right) \cdot \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(-1 \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + -1 \cdot \frac{-1 \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot i\right)\right) + \frac{1}{3} \cdot \frac{{i}^{2}}{n}}{n}\right) \cdot \left(-1 \cdot n\right)\right)} \]

   8. Simplified 92.0%

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{\frac{\left(i \cdot i\right) \cdot 0.3333333333333333}{n} - i \cdot \mathsf{fma}\left(i, 0.5, 0.5\right)}{-n} - \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right)\right) \cdot \left(-n\right)\right)} \]

   if 3.09999999999999975e117 < i

   1. Initial program 54.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 56.5

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

   4. Applied egg-rr 56.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-log1p.f64 N/A

         \[\leadsto 100 \cdot \frac{e^{n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}} - 1}{\frac{i}{n}} \]

      3. lift-*.f64 N/A

         \[\leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} - 1}{\frac{i}{n}} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}{\color{blue}{\frac{i}{n}}} \]

      5. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{i}, n \cdot {\color{blue}{\left(\frac{i}{n}\right)}}^{n}, \frac{\mathsf{neg}\left(n\right)}{i}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 58.1

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

   9. Simplified 58.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{-95}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right) + \frac{\frac{\left(i \cdot i\right) \cdot 0.3333333333333333}{n} - i \cdot \mathsf{fma}\left(i, 0.5, 0.5\right)}{n}\right)\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+117}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{1}{i}, n \cdot {\left(\frac{i}{n}\right)}^{n}, \frac{n}{-i}\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 33.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024218 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))