Compound Interest

Percentage Accurate: 28.5% → 95.8%

Time: 9.0s

Alternatives: 8

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    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: 28.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    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: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -1e-11)
     t_0
     (if (<= t_0 0.0)
       (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
       (if (<= t_0 INFINITY)
         t_0
         (fma 100.0 n (* 100.0 (* i (* n (- 0.5 (* 0.5 (pow n -1.0))))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(100.0, n, (100.0 * (i * (n * (0.5 - (0.5 * pow(n, -1.0)))))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -1e-11)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = fma(100.0, n, Float64(100.0 * Float64(i * Float64(n * Float64(0.5 - Float64(0.5 * (n ^ -1.0)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e-11], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n + N[(100.0 * N[(i * N[(n * N[(0.5 - N[(0.5 * N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -9.99999999999999939e-12 or -0.0 < (*.f64 #s(literal 100 binary64) (/.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 97.3%

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

   if -9.99999999999999939e-12 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if +inf.0 < (*.f64 #s(literal 100 binary64) (/.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. inv-pow N/A

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

      8. lower-pow.f64 88.0

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right) \]

   7. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (*
        100.0
        (fma
         (* n i)
         (-
          (fma 0.4583333333333333 (pow n -2.0) 0.041666666666666664)
          (fma (pow n -3.0) 0.25 (/ 0.25 n)))
         (*
          (-
           (fma (pow n -2.0) 0.3333333333333333 0.16666666666666666)
           (/ 0.5 n))
          n)))
       i
       (* (* (- 0.5 (/ 0.5 n)) n) 100.0))
      i
      (* n 100.0))
     (if (<= t_0 0.0)
       (* (/ (expm1 (* (log1p (/ i n)) n)) (/ i n)) 100.0)
       (if (<= t_0 INFINITY)
         t_0
         (fma 100.0 n (* 100.0 (* i (* n (- 0.5 (* 0.5 (pow n -1.0))))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((100.0 * fma((n * i), (fma(0.4583333333333333, pow(n, -2.0), 0.041666666666666664) - fma(pow(n, -3.0), 0.25, (0.25 / n))), ((fma(pow(n, -2.0), 0.3333333333333333, 0.16666666666666666) - (0.5 / n)) * n))), i, (((0.5 - (0.5 / n)) * n) * 100.0)), i, (n * 100.0));
	} else if (t_0 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(100.0, n, (100.0 * (i * (n * (0.5 - (0.5 * pow(n, -1.0)))))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(Float64(100.0 * fma(Float64(n * i), Float64(fma(0.4583333333333333, (n ^ -2.0), 0.041666666666666664) - fma((n ^ -3.0), 0.25, Float64(0.25 / n))), Float64(Float64(fma((n ^ -2.0), 0.3333333333333333, 0.16666666666666666) - Float64(0.5 / n)) * n))), i, Float64(Float64(Float64(0.5 - Float64(0.5 / n)) * n) * 100.0)), i, Float64(n * 100.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / Float64(i / n)) * 100.0);
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = fma(100.0, n, Float64(100.0 * Float64(i * Float64(n * Float64(0.5 - Float64(0.5 * (n ^ -1.0)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(100.0 * N[(N[(n * i), $MachinePrecision] * N[(N[(0.4583333333333333 * N[Power[n, -2.0], $MachinePrecision] + 0.041666666666666664), $MachinePrecision] - N[(N[Power[n, -3.0], $MachinePrecision] * 0.25 + N[(0.25 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[n, -2.0], $MachinePrecision] * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n + N[(100.0 * N[(i * N[(n * N[(0.5 - N[(0.5 * N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.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 100.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)} \]

   if -inf.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -0.0 < (*.f64 #s(literal 100 binary64) (/.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 96.2%

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

   if +inf.0 < (*.f64 #s(literal 100 binary64) (/.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. inv-pow N/A

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

      8. lower-pow.f64 88.0

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right) \]

   7. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (*
        100.0
        (fma
         (* n i)
         (-
          (fma 0.4583333333333333 (pow n -2.0) 0.041666666666666664)
          (fma (pow n -3.0) 0.25 (/ 0.25 n)))
         (*
          (-
           (fma (pow n -2.0) 0.3333333333333333 0.16666666666666666)
           (/ 0.5 n))
          n)))
       i
       (* (* (- 0.5 (/ 0.5 n)) n) 100.0))
      i
      (* n 100.0))
     (if (<= t_0 0.0)
       (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
       (if (<= t_0 INFINITY)
         t_0
         (fma 100.0 n (* 100.0 (* i (* n (- 0.5 (* 0.5 (pow n -1.0))))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((100.0 * fma((n * i), (fma(0.4583333333333333, pow(n, -2.0), 0.041666666666666664) - fma(pow(n, -3.0), 0.25, (0.25 / n))), ((fma(pow(n, -2.0), 0.3333333333333333, 0.16666666666666666) - (0.5 / n)) * n))), i, (((0.5 - (0.5 / n)) * n) * 100.0)), i, (n * 100.0));
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(100.0, n, (100.0 * (i * (n * (0.5 - (0.5 * pow(n, -1.0)))))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(Float64(100.0 * fma(Float64(n * i), Float64(fma(0.4583333333333333, (n ^ -2.0), 0.041666666666666664) - fma((n ^ -3.0), 0.25, Float64(0.25 / n))), Float64(Float64(fma((n ^ -2.0), 0.3333333333333333, 0.16666666666666666) - Float64(0.5 / n)) * n))), i, Float64(Float64(Float64(0.5 - Float64(0.5 / n)) * n) * 100.0)), i, Float64(n * 100.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = fma(100.0, n, Float64(100.0 * Float64(i * Float64(n * Float64(0.5 - Float64(0.5 * (n ^ -1.0)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(100.0 * N[(N[(n * i), $MachinePrecision] * N[(N[(0.4583333333333333 * N[Power[n, -2.0], $MachinePrecision] + 0.041666666666666664), $MachinePrecision] - N[(N[Power[n, -3.0], $MachinePrecision] * 0.25 + N[(0.25 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[n, -2.0], $MachinePrecision] * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n + N[(100.0 * N[(i * N[(n * N[(0.5 - N[(0.5 * N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.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 100.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)} \]

   if -inf.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 27.8%

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

      1. lift-/.f64 N/A

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

      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. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if -0.0 < (*.f64 #s(literal 100 binary64) (/.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 96.2%

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

   if +inf.0 < (*.f64 #s(literal 100 binary64) (/.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. inv-pow N/A

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

      8. lower-pow.f64 88.0

         \[\leadsto \mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right) \]

   7. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot {n}^{-1}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)\\ t_1 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(\log \left({\left(\frac{i}{n}\right)}^{n}\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (*
            100.0
            (fma
             (* n i)
             (-
              (fma 0.4583333333333333 (pow n -2.0) 0.041666666666666664)
              (fma (pow n -3.0) 0.25 (/ 0.25 n)))
             (*
              (-
               (fma (pow n -2.0) 0.3333333333333333 0.16666666666666666)
               (/ 0.5 n))
              n)))
           i
           (* (* (- 0.5 (/ 0.5 n)) n) 100.0))
          i
          (* n 100.0)))
        (t_1 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 0.0)
       (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
       (if (<= t_1 1000.0)
         (* 100.0 (/ (* n (expm1 (log (pow (/ i n) n)))) i))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = fma(fma((100.0 * fma((n * i), (fma(0.4583333333333333, pow(n, -2.0), 0.041666666666666664) - fma(pow(n, -3.0), 0.25, (0.25 / n))), ((fma(pow(n, -2.0), 0.3333333333333333, 0.16666666666666666) - (0.5 / n)) * n))), i, (((0.5 - (0.5 / n)) * n) * 100.0)), i, (n * 100.0));
	double t_1 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_1 <= 1000.0) {
		tmp = 100.0 * ((n * expm1(log(pow((i / n), n)))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(fma(Float64(100.0 * fma(Float64(n * i), Float64(fma(0.4583333333333333, (n ^ -2.0), 0.041666666666666664) - fma((n ^ -3.0), 0.25, Float64(0.25 / n))), Float64(Float64(fma((n ^ -2.0), 0.3333333333333333, 0.16666666666666666) - Float64(0.5 / n)) * n))), i, Float64(Float64(Float64(0.5 - Float64(0.5 / n)) * n) * 100.0)), i, Float64(n * 100.0))
	t_1 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_1 <= 1000.0)
		tmp = Float64(100.0 * Float64(Float64(n * expm1(log((Float64(i / n) ^ n)))) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(100.0 * N[(N[(n * i), $MachinePrecision] * N[(N[(0.4583333333333333 * N[Power[n, -2.0], $MachinePrecision] + 0.041666666666666664), $MachinePrecision] - N[(N[Power[n, -3.0], $MachinePrecision] * 0.25 + N[(0.25 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[n, -2.0], $MachinePrecision] * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(100.0 * N[(N[(n * N[(Exp[N[Log[N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -inf.0 or 1e3 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)))

   1. Initial program 21.2%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.4583333333333333, {n}^{-2}, 0.041666666666666664\right) - \mathsf{fma}\left({n}^{-3}, 0.25, \frac{0.25}{n}\right), \left(\mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100\right), i, n \cdot 100\right)} \]

   if -inf.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 27.8%

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

      1. lift-/.f64 N/A

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

      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. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 1e3

   1. Initial program 94.3%

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

   3. Taylor expanded in i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      5. diff-log N/A

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

      6. lower-log.f64 N/A

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

      7. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

   10. Applied rewrites 93.8%

       \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \mathsf{expm1}\left(\log \left({\left(\frac{i}{n}\right)}^{n}\right)\right)}{i}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8.6 \cdot 10^{+229} \lor \neg \left(i \leq 700\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(\log \left({\left(\frac{i}{n}\right)}^{n}\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -8.6e+229) (not (<= i 700.0)))
   (* 100.0 (/ (* n (expm1 (log (pow (/ i n) n)))) i))
   (* (fma (/ (expm1 i) i) 100.0 (* (/ (* (exp i) i) n) -50.0)) n)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -8.6e+229) || !(i <= 700.0)) {
		tmp = 100.0 * ((n * expm1(log(pow((i / n), n)))) / i);
	} else {
		tmp = fma((expm1(i) / i), 100.0, (((exp(i) * i) / n) * -50.0)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -8.6e+229) || !(i <= 700.0))
		tmp = Float64(100.0 * Float64(Float64(n * expm1(log((Float64(i / n) ^ n)))) / i));
	else
		tmp = Float64(fma(Float64(expm1(i) / i), 100.0, Float64(Float64(Float64(exp(i) * i) / n) * -50.0)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -8.6e+229], N[Not[LessEqual[i, 700.0]], $MachinePrecision]], N[(100.0 * N[(N[(n * N[(Exp[N[Log[N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0 + N[(N[(N[(N[Exp[i], $MachinePrecision] * i), $MachinePrecision] / n), $MachinePrecision] * -50.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -8.59999999999999982e229 or 700 < i

   1. Initial program 54.9%

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

   3. Taylor expanded in i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      5. diff-log N/A

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

      6. lower-log.f64 N/A

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

      7. lift-/.f64 39.2

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

   8. Applied rewrites 39.2%

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

   10. Applied rewrites 65.1%

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

   if -8.59999999999999982e229 < i < 700

   1. Initial program 20.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right) \cdot \color{blue}{n} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right) \cdot \color{blue}{n} \]

      3. +-commutative N/A

         \[\leadsto \left(100 \cdot \frac{e^{i} - 1}{i} + -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{e^{i} - 1}{i} \cdot 100 + -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      7. lower-expm1.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

      13. lower-exp.f64 85.5

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.6 \cdot 10^{+229} \lor \neg \left(i \leq 700\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(\log \left({\left(\frac{i}{n}\right)}^{n}\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ \mathbf{if}\;i \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{t\_0}^{2}}{i}, 0.5, {i}^{-2}\right), n, \frac{t\_0}{i}\right)\right) \cdot \left(n \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (log (/ i n))))
   (if (<= i 1.05e+27)
     (* (fma (/ (expm1 i) i) 100.0 (* (/ (* (exp i) i) n) -50.0)) n)
     (*
      (* 100.0 (fma (fma (/ (pow t_0 2.0) i) 0.5 (pow i -2.0)) n (/ t_0 i)))
      (* n n)))))

C

double code(double i, double n) {
	double t_0 = log((i / n));
	double tmp;
	if (i <= 1.05e+27) {
		tmp = fma((expm1(i) / i), 100.0, (((exp(i) * i) / n) * -50.0)) * n;
	} else {
		tmp = (100.0 * fma(fma((pow(t_0, 2.0) / i), 0.5, pow(i, -2.0)), n, (t_0 / i))) * (n * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = log(Float64(i / n))
	tmp = 0.0
	if (i <= 1.05e+27)
		tmp = Float64(fma(Float64(expm1(i) / i), 100.0, Float64(Float64(Float64(exp(i) * i) / n) * -50.0)) * n);
	else
		tmp = Float64(Float64(100.0 * fma(fma(Float64((t_0 ^ 2.0) / i), 0.5, (i ^ -2.0)), n, Float64(t_0 / i))) * Float64(n * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[i, 1.05e+27], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0 + N[(N[(N[(N[Exp[i], $MachinePrecision] * i), $MachinePrecision] / n), $MachinePrecision] * -50.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(100.0 * N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / i), $MachinePrecision] * 0.5 + N[Power[i, -2.0], $MachinePrecision]), $MachinePrecision] * n + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 1.04999999999999997e27

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right) \cdot \color{blue}{n} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right) \cdot \color{blue}{n} \]

      3. +-commutative N/A

         \[\leadsto \left(100 \cdot \frac{e^{i} - 1}{i} + -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{e^{i} - 1}{i} \cdot 100 + -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      7. lower-expm1.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{i \cdot e^{i}}{n} \cdot -50\right) \cdot n \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

      13. lower-exp.f64 82.9

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{expm1}\left(i\right)}{i}, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n} \]

   if 1.04999999999999997e27 < i

   1. Initial program 47.5%

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

   3. Taylor expanded in i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      5. diff-log N/A

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

      6. lower-log.f64 N/A

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

      7. lift-/.f64 30.4

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

   8. Applied rewrites 30.4%

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in n around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 44.8%

       \[\leadsto \color{blue}{\left(100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\log \left(\frac{i}{n}\right)}^{2}}{i}, 0.5, {i}^{-2}\right), n, \frac{\log \left(\frac{i}{n}\right)}{i}\right)\right) \cdot \left(n \cdot n\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 23.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{i}{n}\right)\\ t_1 := \mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right)\\ \mathbf{if}\;i \leq -3.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(--100\right) \cdot \mathsf{fma}\left(\mathsf{expm1}\left(t\_1 \cdot n\right), n, {n}^{3} \cdot \frac{{\left(e^{n}\right)}^{t\_1}}{i}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{t\_0}^{2}}{i}, 0.5, {i}^{-2}\right), n, \frac{t\_0}{i}\right)\right) \cdot \left(n \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (log (/ i n))) (t_1 (fma (log (/ -1.0 i)) -1.0 (log (/ -1.0 n)))))
   (if (<= i -3.2e+36)
     (/
      (*
       (- -100.0)
       (fma (expm1 (* t_1 n)) n (* (pow n 3.0) (/ (pow (exp n) t_1) i))))
      i)
     (*
      (* 100.0 (fma (fma (/ (pow t_0 2.0) i) 0.5 (pow i -2.0)) n (/ t_0 i)))
      (* n n)))))

C

double code(double i, double n) {
	double t_0 = log((i / n));
	double t_1 = fma(log((-1.0 / i)), -1.0, log((-1.0 / n)));
	double tmp;
	if (i <= -3.2e+36) {
		tmp = (-(-100.0) * fma(expm1((t_1 * n)), n, (pow(n, 3.0) * (pow(exp(n), t_1) / i)))) / i;
	} else {
		tmp = (100.0 * fma(fma((pow(t_0, 2.0) / i), 0.5, pow(i, -2.0)), n, (t_0 / i))) * (n * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = log(Float64(i / n))
	t_1 = fma(log(Float64(-1.0 / i)), -1.0, log(Float64(-1.0 / n)))
	tmp = 0.0
	if (i <= -3.2e+36)
		tmp = Float64(Float64(Float64(-(-100.0)) * fma(expm1(Float64(t_1 * n)), n, Float64((n ^ 3.0) * Float64((exp(n) ^ t_1) / i)))) / i);
	else
		tmp = Float64(Float64(100.0 * fma(fma(Float64((t_0 ^ 2.0) / i), 0.5, (i ^ -2.0)), n, Float64(t_0 / i))) * Float64(n * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision] * -1.0 + N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.2e+36], N[(N[((--100.0) * N[(N[(Exp[N[(t$95$1 * n), $MachinePrecision]] - 1), $MachinePrecision] * n + N[(N[Power[n, 3.0], $MachinePrecision] * N[(N[Power[N[Exp[n], $MachinePrecision], t$95$1], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(N[(100.0 * N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / i), $MachinePrecision] * 0.5 + N[Power[i, -2.0], $MachinePrecision]), $MachinePrecision] * n + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < -3.1999999999999999e36

   1. Initial program 69.7%

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

   3. Taylor expanded in i around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i} \]

   5. Applied rewrites 58.9%

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

   if -3.1999999999999999e36 < i

   1. Initial program 17.0%

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

   3. Taylor expanded in i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 13.4%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{i} \]

      5. diff-log N/A

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

      6. lower-log.f64 N/A

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

      7. lift-/.f64 15.3

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

   8. Applied rewrites 15.3%

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

   10. Applied rewrites 22.1%

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

   11. Taylor expanded in n around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 18.0%

       \[\leadsto \color{blue}{\left(100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\log \left(\frac{i}{n}\right)}^{2}}{i}, 0.5, {i}^{-2}\right), n, \frac{\log \left(\frac{i}{n}\right)}{i}\right)\right) \cdot \left(n \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(--100\right) \cdot \mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right) \cdot n\right), n, {n}^{3} \cdot \frac{{\left(e^{n}\right)}^{\left(\mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right)\right)}}{i}\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\log \left(\frac{i}{n}\right)}^{2}}{i}, 0.5, {i}^{-2}\right), n, \frac{\log \left(\frac{i}{n}\right)}{i}\right)\right) \cdot \left(n \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 13.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right)\\ \frac{\left(--100\right) \cdot \mathsf{fma}\left(\mathsf{expm1}\left(t\_0 \cdot n\right), n, {n}^{3} \cdot \frac{{\left(e^{n}\right)}^{t\_0}}{i}\right)}{i} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (fma (log (/ -1.0 i)) -1.0 (log (/ -1.0 n)))))
   (/
    (*
     (- -100.0)
     (fma (expm1 (* t_0 n)) n (* (pow n 3.0) (/ (pow (exp n) t_0) i))))
    i)))

C

double code(double i, double n) {
	double t_0 = fma(log((-1.0 / i)), -1.0, log((-1.0 / n)));
	return (-(-100.0) * fma(expm1((t_0 * n)), n, (pow(n, 3.0) * (pow(exp(n), t_0) / i)))) / i;
}

Julia

function code(i, n)
	t_0 = fma(log(Float64(-1.0 / i)), -1.0, log(Float64(-1.0 / n)))
	return Float64(Float64(Float64(-(-100.0)) * fma(expm1(Float64(t_0 * n)), n, Float64((n ^ 3.0) * Float64((exp(n) ^ t_0) / i)))) / i)
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision] * -1.0 + N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[((--100.0) * N[(N[(Exp[N[(t$95$0 * n), $MachinePrecision]] - 1), $MachinePrecision] * n + N[(N[Power[n, 3.0], $MachinePrecision] * N[(N[Power[N[Exp[n], $MachinePrecision], t$95$0], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in i around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i}\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto -\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i} \]

   3. lower-/.f64 N/A

      \[\leadsto -\frac{-100 \cdot \left(n \cdot \left(e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)} - 1\right)\right) + -100 \cdot \frac{{n}^{3} \cdot e^{n \cdot \left(\log \left(\mathsf{neg}\left(\frac{1}{n}\right)\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}}{i}}{i} \]

5. Applied rewrites 17.6%

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

6. Final simplification 17.6%

   \[\leadsto \frac{\left(--100\right) \cdot \mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right) \cdot n\right), n, {n}^{3} \cdot \frac{{\left(e^{n}\right)}^{\left(\mathsf{fma}\left(\log \left(\frac{-1}{i}\right), -1, \log \left(\frac{-1}{n}\right)\right)\right)}}{i}\right)}{i} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))